100 results found for "subsets" in all classes.
\(^{n}C_0+^{n}C_1+\cdots+^{n}C_n\) को subsets से जोड़ने पर कुल कितने subsets मिलते हैं?
Connecting \(^{n}C_0+^{n}C_1+\cdots+^{n}C_n\) with subsets gives how many total subsets?
#subsets
#binomial_sum
#combination_connection
A (n!)
B \(2^n\)
C \(^{n}P_n\)
D \(n^2\)
Explanation opens after your attempt
Correct Answer
B. \(2^n\)
Step 1
Concept
Each object has two options include or exclude. In exams remember \(2^n\) for total subsets.
Step 2
Why this answer is correct
The correct answer is B. \(2^n\). Each object has two options include or exclude. In exams remember \(2^n\) for total subsets.
Step 3
Exam Tip
हर वस्तु के लिए शामिल या अलग दो विकल्प होते हैं। परीक्षा में total subsets के लिए \(2^n\) याद रखें।
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यदि किसी समुच्चय के कुल उपसमुच्चय (32) हैं तो उस समुच्चय के उचित उपसमुच्चय कितने होंगे?
If a set has (32) total subsets then how many proper subsets will it have?
#sets
#proper subsets
#counting
A (30)
B (31)
C (32)
D (33)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^n=32\) so proper subsets are (32-1=31). In exams do not count the original set as a proper subset.
Step 2
Why this answer is correct
The correct answer is B. (31). Total subsets are \(2^n=32\) so proper subsets are (32-1=31). In exams do not count the original set as a proper subset.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^n=32\) हैं इसलिए उचित उपसमुच्चय (32-1=31) होंगे। परीक्षा में मूल समुच्चय को उचित उपसमुच्चय में न गिनें।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5\}\) हैं तो (B) के कितने उपसमुच्चय (A) के भी उपसमुच्चय हैं?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5\}\), how many subsets of (B) are also subsets of (A)?
#sets
#subsets
#common_subsets
A (3)
B (6)
C (8)
D (32)
Explanation opens after your attempt
Step 1
Concept
The subsets lying inside (A) are subsets of both, and their number is \(2^3=8\). Decide the common limit using the smaller set.
Step 2
Why this answer is correct
The correct answer is C. (8). The subsets lying inside (A) are subsets of both, and their number is \(2^3=8\). Decide the common limit using the smaller set.
Step 3
Exam Tip
जो उपसमुच्चय (A) के अंदर हैं वही दोनों के लिए उपसमुच्चय होंगे और उनकी संख्या \(2^3=8\) है। साझा सीमा को छोटे समुच्चय से तय करें।
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यदि \(A=\{p,q,r,s\}\) और \(B=\{p,r\}\) हैं तो (B) के कौन से उपसमुच्चय (A) के भी उपसमुच्चय हैं?
If \(A=\{p,q,r,s\}\) and \(B=\{p,r\}\), which subsets of (B) are also subsets of (A)?
#sets
#subset_transitivity
#reasoning
A केवल ({p,r}) / Only ({p,r})
B केवल \(\varnothing\) / Only \(\varnothing\)
C (B) के सभी उपसमुच्चय / All subsets of (B)
D केवल एकल अवयव वाले उपसमुच्चय / Only singleton subsets
Explanation opens after your attempt
Correct Answer
C. (B) के सभी उपसमुच्चय / All subsets of (B)
Step 1
Concept
Because \(B\subset A\), every subset of (B) is also a subset of (A). This uses transitive reasoning.
Step 2
Why this answer is correct
The correct answer is C. (B) के सभी उपसमुच्चय / All subsets of (B). Because \(B\subset A\), every subset of (B) is also a subset of (A). This uses transitive reasoning.
Step 3
Exam Tip
क्योंकि \(B\subset A\), इसलिए (B) का हर उपसमुच्चय (A) का भी उपसमुच्चय है। यह ट्रांजिटिव सोच का प्रयोग है।
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यदि \(A=\{a,b,c,d,e\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी होगी जिनमें (a) और (b) दोनों अवश्य हों?
If \(A=\{a,b,c,d,e\}\) then how many subsets contain both (a) and (b) necessarily?
#sets
#subsets
#counting
#conditional-subsets
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
After fixing (a) and (b), the remaining (3) elements can be chosen freely. So the number is \(2^3=8\) and fixed elements should be separated in such questions.
Step 2
Why this answer is correct
The correct answer is C. (8). After fixing (a) and (b), the remaining (3) elements can be chosen freely. So the number is \(2^3=8\) and fixed elements should be separated in such questions.
Step 3
Exam Tip
(a) और (b) को निश्चित रखने पर बाकी (3) अवयव स्वतंत्र चुने जाते हैं। इसलिए संख्या \(2^3=8\) है और ऐसे प्रश्नों में निश्चित अवयव अलग कर दें।
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समुच्चय \(A=\{1,2,3\}\) के सभी एक-सदस्यीय उपसमुच्चय कौन-से हैं?
What are all the one-element subsets of \(A=\{1,2,3\}\)?
#singleton-subsets
#subsets
#sets
A ({1},{2},{3})
B ({1,2},{2,3})
C \(\varnothing,{1}\)
D ({1,2,3}) केवल / only ({1,2,3})
Explanation opens after your attempt
Correct Answer
A. ({1},{2},{3})
Step 1
Concept
A one-element subset contains exactly one element.
Step 2
Why this answer is correct
Since (A) has three elements, the one-element subsets are ({1},{2},{3}).
Step 3
Exam Tip
Put elements inside braces when writing subsets. चरण 1: एक-सदस्यीय उपसमुच्चय में केवल एक सदस्य होगा। चरण 2: (A) के तीन सदस्य हैं, इसलिए ({1},{2},{3}) मिलेंगे। चरण 3: उपसमुच्चय बनाते समय सदस्यों को कोष्ठकों में रखें।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (2) को शामिल करने और (4) को शामिल न करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets contain (2) and do not contain (4)?
#subsets
#counting
#conditions
A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
Element (2) is fixed, (4) is excluded, and (1,3,5) are free. Hence \(2^3=8\) subsets are possible.
Step 2
Why this answer is correct
The correct answer is B. (8). Element (2) is fixed, (4) is excluded, and (1,3,5) are free. Hence \(2^3=8\) subsets are possible.
Step 3
Exam Tip
(2) निश्चित है, (4) बाहर है और (1,3,5) स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय बनेंगे।
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यदि \(A=\{1,2,3,4\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें सदस्यों का योग सम हो?
If \(A=\{1,2,3,4\}\), how many subsets have an even sum of elements?
#subsets
#counting
#parity
A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
For an even sum, the number of selected odd elements must be even. Among the (16) subsets, half, that is (8), give even sum.
Step 2
Why this answer is correct
The correct answer is C. (8). For an even sum, the number of selected odd elements must be even. Among the (16) subsets, half, that is (8), give even sum.
Step 3
Exam Tip
सम योग के लिए चुने गए विषम सदस्यों की संख्या सम होनी चाहिए। चार-सदस्यीय सेट में कुल (16) उपसमुच्चयों में आधे यानी (8) सम योग देते हैं।
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यदि \(A=\{1,2,3,4\}\), तो (2) और (3) को साथ-साथ शामिल या साथ-साथ बाहर रखने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets either contain both (2,3) or contain neither of them?
#subsets
#counting
#paired-condition
A (4)
B (6)
C (8)
D (12)
Explanation opens after your attempt
Step 1
Concept
There are two valid states for (2,3), and (1,4) are free. Hence \(2\times2^2=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). There are two valid states for (2,3), and (1,4) are free. Hence \(2\times2^2=8\).
Step 3
Exam Tip
(2,3) के लिए दो वैध स्थितियां हैं और (1,4) स्वतंत्र हैं। इसलिए \(2\times2^2=8\) है।
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यदि \(A=\{1,2,3,4,5\}\), तो कम से कम (4) सदस्य वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets have at least (4) elements?
#subsets
#counting
#at-least
A (5)
B (6)
C (10)
D (16)
Explanation opens after your attempt
Step 1
Concept
There are \(\binom{5}{4}=5\) four-element subsets and (1) five-element subset. Total is (5+1=6).
Step 2
Why this answer is correct
The correct answer is B. (6). There are \(\binom{5}{4}=5\) four-element subsets and (1) five-element subset. Total is (5+1=6).
Step 3
Exam Tip
चार-सदस्यीय उपसमुच्चय \(\binom{5}{4}=5\) और पांच-सदस्यीय (1) है। कुल (5+1=6) है।
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यदि \(A=\{1,2,3,4,5,6\}\), तो ठीक दो विषम सदस्यों वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets have exactly two odd elements?
#subsets
#counting
#odd-elements
A (3)
B (6)
C (12)
D (24)
Explanation opens after your attempt
Step 1
Concept
Choose two from the three odd numbers, and the three even numbers are optional. The count is \(\binom{3}{2}\times2^3=24\).
Step 2
Why this answer is correct
The correct answer is D. (24). Choose two from the three odd numbers, and the three even numbers are optional. The count is \(\binom{3}{2}\times2^3=24\).
Step 3
Exam Tip
तीन विषम संख्याओं में से दो चुनें और तीन सम संख्याएँ वैकल्पिक हैं। संख्या \(\binom{3}{2}\times2^3=24\) है।
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यदि \(A=\{1,2,3,4\}\), तो केवल सम सदस्यों से बने उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets can be formed using only even elements?
#subsets
#counting
#even-elements
A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
The even elements are (2,4). The number of their subsets is \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is C. (4). The even elements are (2,4). The number of their subsets is \(2^2=4\).
Step 3
Exam Tip
सम सदस्य (2,4) हैं। इनके सभी उपसमुच्चयों की संख्या \(2^2=4\) है।
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यदि \(A=\{a,b,c,d\}\), तो (a) या (b) में से कम से कम एक को शामिल करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{a,b,c,d\}\), how many subsets contain at least one of (a) or (b)?
#subsets
#counting
#at-least-one
A (8)
B (10)
C (12)
D (14)
Explanation opens after your attempt
Step 1
Concept
Total subsets are (16), and those excluding both (a,b) are \(2^2=4\). Hence (16-4=12).
Step 2
Why this answer is correct
The correct answer is C. (12). Total subsets are (16), and those excluding both (a,b) are \(2^2=4\). Hence (16-4=12).
Step 3
Exam Tip
कुल उपसमुच्चय (16) हैं और (a,b) दोनों को बाहर रखने वाले \(2^2=4\) हैं। इसलिए (16-4=12) है।
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यदि \(A=\{1,2,3,4,5\}\), तो (1) को शामिल और (5) को बाहर रखने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets contain (1) and exclude (5)?
#subsets
#counting
#conditions
A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
Element (1) is fixed, (5) is forbidden, and (2,3,4) are free. Hence \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). Element (1) is fixed, (5) is forbidden, and (2,3,4) are free. Hence \(2^3=8\).
Step 3
Exam Tip
(1) निश्चित है, (5) निषिद्ध है और (2,3,4) स्वतंत्र हैं। इसलिए \(2^3=8\) है।
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यदि (A) में (5) सदस्य हैं, तो (A) के तीन-सदस्यीय उपसमुच्चयों की संख्या कितनी है?
If (A) has (5) elements, how many three-element subsets does (A) have?
#subsets
#combinations
#counting
A (5)
B (10)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose three elements is \(\binom{5}{3}=10\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose three elements is \(\binom{5}{3}=10\). Order is not counted in subsets.
Step 3
Exam Tip
तीन सदस्य चुनने की संख्या \(\binom{5}{3}=10\) है। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A=\{0,1,2,3\}\), तो (0) को शामिल करने वाले ठीक दो-सदस्यीय उपसमुच्चयों की संख्या कितनी है?
If \(A=\{0,1,2,3\}\), how many two-element subsets contain (0)?
#subsets
#counting
#two-element
A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
Element (0) is fixed, and the second element is chosen from (1,2,3). So there are (3) subsets.
Step 2
Why this answer is correct
The correct answer is B. (3). Element (0) is fixed, and the second element is chosen from (1,2,3). So there are (3) subsets.
Step 3
Exam Tip
(0) निश्चित है और दूसरा सदस्य (1,2,3) में से चुना जाएगा। इसलिए (3) उपसमुच्चय बनते हैं।
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यदि \(A=\{1,2,3,4\}\) है, तो कम से कम एक सदस्य वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets have at least one element?
#subsets
#counting
#non-empty
A (14)
B (15)
C (16)
D (8)
Explanation opens after your attempt
Step 1
Concept
Total subsets are (16), and only the empty set is excluded. Hence (16-1=15).
Step 2
Why this answer is correct
The correct answer is B. (15). Total subsets are (16), and only the empty set is excluded. Hence (16-1=15).
Step 3
Exam Tip
कुल उपसमुच्चय (16) हैं और केवल रिक्त समुच्चय हटाना है। इसलिए (16-1=15) है।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो केवल विषम सदस्यों से बने उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets can be formed using only odd elements?
#subsets
#counting
#odd-elements
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
The odd elements are (1,3,5). Their subsets are \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). The odd elements are (1,3,5). Their subsets are \(2^3=8\).
Step 3
Exam Tip
विषम सदस्य (1,3,5) हैं। इनके उपसमुच्चय \(2^3=8\) होंगे।
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यदि \(A=\{2,3,5,7\}\) है, तो कितने उपसमुच्चय (2) और (7) दोनों को शामिल करते हैं?
If \(A=\{2,3,5,7\}\), how many subsets contain both (2) and (7)?
#subsets
#counting
#fixed-elements
A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Elements (2,7) are fixed, while (3,5) are optional. So there are \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. (4). Elements (2,7) are fixed, while (3,5) are optional. So there are \(2^2=4\) subsets.
Step 3
Exam Tip
(2,7) निश्चित हैं और (3,5) वैकल्पिक हैं। इसलिए \(2^2=4\) उपसमुच्चय हैं।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (1,2) को शामिल और (5) को बाहर रखने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\), how many subsets contain (1,2) and exclude (5)?
#subsets
#counting
#conditions
A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Elements (1,2) are fixed, (5) is forbidden, and (3,4) are optional. Hence the answer is \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is B. (4). Elements (1,2) are fixed, (5) is forbidden, and (3,4) are optional. Hence the answer is \(2^2=4\).
Step 3
Exam Tip
(1,2) निश्चित हैं, (5) निषिद्ध है और (3,4) वैकल्पिक हैं। इसलिए \(2^2=4\) उत्तर है।
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यदि \(A=\{p,q,r,s,t\}\) है, तो (p) को न शामिल करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{p,q,r,s,t\}\), how many subsets do not contain (p)?
#subsets
#counting
#excluded-element
A (8)
B (16)
C (31)
D (32)
Explanation opens after your attempt
Step 1
Concept
After excluding (p), (4) elements remain free. Thus \(2^4=16\) subsets are possible.
Step 2
Why this answer is correct
The correct answer is B. (16). After excluding (p), (4) elements remain free. Thus \(2^4=16\) subsets are possible.
Step 3
Exam Tip
(p) हटाने के बाद (4) सदस्य स्वतंत्र रहते हैं। इसलिए \(2^4=16\) उपसमुच्चय बनते हैं।
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यदि \(A=\{1,2,3,4\}\) है, तो (2) को शामिल करने वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\), how many subsets contain (2)?
#subsets
#counting
#fixed-element
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Element (2) is fixed, and the remaining (3) elements are optional. Therefore the count is \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). Element (2) is fixed, and the remaining (3) elements are optional. Therefore the count is \(2^3=8\).
Step 3
Exam Tip
(2) निश्चित है और बाकी (3) सदस्य वैकल्पिक हैं। इसलिए संख्या \(2^3=8\) है।
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यदि \(A=\emptyset\) है, तो (A) के उपसमुच्चयों की संख्या क्या होगी?
If \(A=\emptyset\), how many subsets does (A) have?
#empty-set
#counting
#subsets
A (0)
B (1)
C (2)
D अनंत / Infinite
Explanation opens after your attempt
Step 1
Concept
The empty set has only one subset, itself. The formula gives \(2^0=1\).
Step 2
Why this answer is correct
The correct answer is B. (1). The empty set has only one subset, itself. The formula gives \(2^0=1\).
Step 3
Exam Tip
रिक्त समुच्चय का केवल एक उपसमुच्चय वही स्वयं है। सूत्र \(2^0=1\) देता है।
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चार सदस्यों वाले समुच्चय के उचित उपसमुच्चयों की संख्या कितनी है?
How many proper subsets does a set with four elements have?
#subsets
#proper-subset
#counting
A (15)
B (16)
C (8)
D (14)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\), and removing the whole set leaves (15) proper subsets. Do not forget to subtract the set itself.
Step 2
Why this answer is correct
The correct answer is A. (15). Total subsets are \(2^4=16\), and removing the whole set leaves (15) proper subsets. Do not forget to subtract the set itself.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं और पूरा सेट हटाने पर (15) उचित उपसमुच्चय बचते हैं। अंतिम उत्तर में पूरा सेट घटाना न भूलें।
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तीन सदस्यों वाले समुच्चय के कुल उपसमुच्चयों की संख्या कितनी होती है?
How many subsets does a set with three elements have?
#subsets
#counting
#power-set
A (6)
B (7)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
A set with (n) elements has \(2^n\) subsets. Here \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). A set with (n) elements has \(2^n\) subsets. Here \(2^3=8\).
Step 3
Exam Tip
(n) सदस्यों वाले सेट के उपसमुच्चय \(2^n\) होते हैं। यहाँ \(2^3=8\) है।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के तीन तत्वों वाले उचित उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4\}\) then how many proper subsets of (A) have exactly three elements?
#sets
#proper subsets
#combinations
A (3)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose three elements is \({}^4C_3=4\) and all are proper subsets. In exams a subset with size less than (n) is proper.
Step 2
Why this answer is correct
The correct answer is B. (4). The number of ways to choose three elements is \({}^4C_3=4\) and all are proper subsets. In exams a subset with size less than (n) is proper.
Step 3
Exam Tip
तीन तत्व चुनने के तरीके \({}^4C_3=4\) हैं और ये सभी उचित उपसमुच्चय हैं। परीक्षा में आकार (n) से छोटा हो तो उपसमुच्चय उचित होता है।
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यदि \(A=\emptyset\) है तो (A) के उपसमुच्चयों की संख्या कितनी है?
If \(A=\emptyset\) then how many subsets does (A) have?
#sets
#empty set
#subsets
A (0)
B (1)
C (2)
D अनंत / Infinite
Explanation opens after your attempt
Step 1
Concept
The empty set has only one subset which is the empty set itself. In exams remember \(2^0=1\).
Step 2
Why this answer is correct
The correct answer is B. (1). The empty set has only one subset which is the empty set itself. In exams remember \(2^0=1\).
Step 3
Exam Tip
रिक्त समुच्चय का केवल एक उपसमुच्चय वही रिक्त समुच्चय है। परीक्षा में \(2^0=1\) याद रखें।
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यदि \(A=\{a,b,c,d\}\) है तो (A) के सभी दो तत्वों वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{a,b,c,d\}\) then how many subsets of (A) have exactly two elements?
#sets
#subsets
#combinations
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose two elements is \({}^4C_2=6\). In exams use combinations for subsets of fixed size.
Step 2
Why this answer is correct
The correct answer is B. (6). The number of ways to choose two elements is \({}^4C_2=6\). In exams use combinations for subsets of fixed size.
Step 3
Exam Tip
दो तत्व चुनने की संख्या \({}^4C_2=6\) है। परीक्षा में निश्चित आकार के उपसमुच्चय के लिए संयोजन का प्रयोग करें।
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यदि \(A=\{a,b,c,d,e\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (a) हो, (c) न हो और ठीक 3 अवयव हों?
If \(A=\{a,b,c,d,e\}\), how many subsets contain (a), do not contain (c), and have exactly 3 elements?
#subsets
#counting
#combination
#level 9
A 2
B 3
C 4
D 6
Explanation opens after your attempt
Step 1
Concept
(a) is fixed and (c) is excluded, so choose 2 from (b,d,e). The number is \(\binom{3}{2}=3\).
Step 2
Why this answer is correct
The correct answer is B. 3. (a) is fixed and (c) is excluded, so choose 2 from (b,d,e). The number is \(\binom{3}{2}=3\).
Step 3
Exam Tip
(a) निश्चित है और (c) हटाया गया है, इसलिए (b,d,e) में से 2 चुनने हैं। संख्या \(\binom{3}{2}=3\) है।
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यदि \(A=\{a,b,c,d\}\), तो (A) के उन उपसमुच्चयों की संख्या कितनी है जिनमें कम से कम एक अवयव हो?
If \(A=\{a,b,c,d\}\), how many subsets of (A) have at least one element?
#nonempty subsets
#counting
A 15
B 16
C 8
D 4
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^4=16\). For at least one element, remove the empty set, giving 15.
Step 2
Why this answer is correct
The correct answer is A. 15. Total subsets are \(2^4=16\). For at least one element, remove the empty set, giving 15.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^4=16\) हैं। कम से कम एक अवयव के लिए रिक्त समुच्चय हटाएं, इसलिए 15।
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यदि \(A=\{1,2,3,4,5\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 अवश्य हो?
If \(A=\{1,2,3,4,5\}\), how many subsets must contain 1?
#subsets
#contains element
#counting
A 8
B 16
C 24
D 32
Explanation opens after your attempt
Step 1
Concept
Fix 1, and each of the remaining 4 elements may be chosen or not. So there are \(2^4=16\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 16. Fix 1, and each of the remaining 4 elements may be chosen or not. So there are \(2^4=16\) subsets.
Step 3
Exam Tip
1 को निश्चित रखें और शेष 4 अवयवों को चुनना या न चुनना स्वतंत्र है। इसलिए \(2^4=16\) उपसमुच्चय मिलते हैं।
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किसी 5 अवयव वाले समुच्चय के कुल उपसमुच्चयों की संख्या क्या होगी?
What is the total number of subsets of a set having 5 elements?
#number of subsets
#power set
#hard
A 10
B 25
C 32
D 120
Explanation opens after your attempt
Step 1
Concept
A set with (n) elements has \(2^n\) subsets. Here \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is C. 32. A set with (n) elements has \(2^n\) subsets. Here \(2^5=32\).
Step 3
Exam Tip
(n) अवयवों वाले समुच्चय के उपसमुच्चय \(2^n\) होते हैं। यहां \(2^5=32\) है।
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यदि \(A=\{1,2,3,4\}\) और \(B=\{1,3\}\) हैं तो (A) के कितने उपसमुच्चय (B) को उपसमुच्चय के रूप में रखते हैं?
If \(A=\{1,2,3,4\}\) and \(B=\{1,3\}\), how many subsets of (A) contain (B) as a subset?
#sets
#subsets
#contain_subset
A (2)
B (4)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
(1) and (3) are fixed, while (2,4) are optional, so there are \(2^2=4\) subsets. When a subset must be contained, fix its elements first.
Step 2
Why this answer is correct
The correct answer is B. (4). (1) and (3) are fixed, while (2,4) are optional, so there are \(2^2=4\) subsets. When a subset must be contained, fix its elements first.
Step 3
Exam Tip
(1) और (3) निश्चित हैं तथा (2,4) स्वतंत्र हैं इसलिए \(2^2=4\) उपसमुच्चय होंगे। किसी उपसमुच्चय को शामिल रखना हो तो उसके अवयव पहले तय करें।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (2) और (3) दोनों हों?
If \(A=\{1,2,3,4\}\), how many subsets of (A) contain both (2) and (3)?
#sets
#subsets
#restricted_counting
A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(2) and (3) are fixed, while (1,4) are optional, so \(2^2=4\) subsets are obtained. Include fixed elements first.
Step 2
Why this answer is correct
The correct answer is B. (4). (2) and (3) are fixed, while (1,4) are optional, so \(2^2=4\) subsets are obtained. Include fixed elements first.
Step 3
Exam Tip
(2) और (3) निश्चित हैं तथा (1,4) स्वतंत्र हैं इसलिए \(2^2=4\) उपसमुच्चय मिलते हैं। निश्चित अवयवों को पहले शामिल करें।
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यदि \(A={x:x\) (100) से छोटे (25) के धनात्मक गुणज हैं(}) तो (A) के उपसमुच्चयों की संख्या क्या है?
If \(A={x:x\) is a positive multiple of (25) less than (100)(}), how many subsets does (A) have?
#sets
#subsets
#multiples
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A=\{25,50,75\}\) has (3) elements, so it has \(2^3=8\) subsets. The boundary (100) is not included.
Step 2
Why this answer is correct
The correct answer is C. (8). \(A=\{25,50,75\}\) has (3) elements, so it has \(2^3=8\) subsets. The boundary (100) is not included.
Step 3
Exam Tip
\(A=\{25,50,75\}\) में (3) अवयव हैं इसलिए \(2^3=8\) उपसमुच्चय होंगे। सीमा में (100) शामिल नहीं है।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (1) हो और (4) न हो?
If \(A=\{1,2,3,4\}\), how many subsets of (A) contain (1) and do not contain (4)?
#sets
#subsets
#restricted_counting
A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (4) is forbidden, so (2,3) are free. Hence \(2^2=4\) subsets are possible.
Step 2
Why this answer is correct
The correct answer is B. (4). (1) is fixed and (4) is forbidden, so (2,3) are free. Hence \(2^2=4\) subsets are possible.
Step 3
Exam Tip
(1) निश्चित और (4) निषिद्ध है इसलिए (2,3) स्वतंत्र हैं। अतः \(2^2=4\) उपसमुच्चय बनते हैं।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के तीन या अधिक अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2,3,4,5\}\), how many subsets of (A) have three or more elements?
#sets
#subsets
#combinations
A (10)
B (16)
C (20)
D (26)
Explanation opens after your attempt
Step 1
Concept
The number is \(\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=16\). For size based counting, add the combinations.
Step 2
Why this answer is correct
The correct answer is B. (16). The number is \(\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=16\). For size based counting, add the combinations.
Step 3
Exam Tip
तीन या अधिक अवयवों की संख्या \(\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=16\) है। आकार आधारित गिनती में संयोजन जोड़ें।
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यदि \(A=\{p,q,r,s\}\) है तो (A) के दो अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{p,q,r,s\}\), how many subsets of (A) have exactly two elements?
#sets
#subsets
#combinations
A (4)
B (6)
C (8)
D (12)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose two elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (6). The number of ways to choose two elements is \(\binom{4}{2}=6\). Order is not counted in subsets.
Step 3
Exam Tip
दो अवयव चुनने के तरीके \(\binom{4}{2}=6\) हैं। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A=\{a,b,c,d,e\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (a) न हो?
If \(A=\{a,b,c,d,e\}\), how many subsets of (A) do not contain (a)?
#sets
#subsets
#counting_without_element
A (8)
B (16)
C (24)
D (31)
Explanation opens after your attempt
Step 1
Concept
After excluding (a), four elements remain and they form \(2^4=16\) subsets. Remove the forbidden element first.
Step 2
Why this answer is correct
The correct answer is B. (16). After excluding (a), four elements remain and they form \(2^4=16\) subsets. Remove the forbidden element first.
Step 3
Exam Tip
(a) हटाने के बाद चार अवयव बचते हैं और उनसे \(2^4=16\) उपसमुच्चय बनते हैं। निषिद्ध अवयव को पहले बाहर करें।
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यदि \(A=\{0,1,2,3\}\) है तो (A) के ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (0) अवश्य हो?
If \(A=\{0,1,2,3\}\), how many subsets of (A) must contain (0)?
#sets
#subsets
#counting
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Fixing (0), the remaining three elements are optional, so there are \(2^3=8\) subsets. For a fixed element, count choices for the remaining elements.
Step 2
Why this answer is correct
The correct answer is C. (8). Fixing (0), the remaining three elements are optional, so there are \(2^3=8\) subsets. For a fixed element, count choices for the remaining elements.
Step 3
Exam Tip
(0) को निश्चित रखने पर बाकी तीन अवयव स्वतंत्र हैं इसलिए \(2^3=8\) उपसमुच्चय मिलते हैं। निश्चित अवयव को चुनकर बाकी पर गिनती करें।
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यदि \(A=\{a,b,c,d\}\) है तो (A) के कम से कम दो अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{a,b,c,d\}\), how many subsets of (A) have at least two elements?
#sets
#subsets
#counting
A (6)
B (10)
C (11)
D (12)
Explanation opens after your attempt
Step 1
Concept
There are (16) total subsets and (1+4=5) subsets with zero or one element. So the answer is (16-5=11).
Step 2
Why this answer is correct
The correct answer is C. (11). There are (16) total subsets and (1+4=5) subsets with zero or one element. So the answer is (16-5=11).
Step 3
Exam Tip
कुल उपसमुच्चय (16) हैं और शून्य या एक अवयव वाले (1+4=5) हैं। इसलिए उत्तर (16-5=11) है।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के दो अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2,3,4,5\}\), how many subsets of (A) have exactly two elements?
#sets
#subsets
#combinations
A (5)
B (10)
C (20)
D (25)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose two elements is \(\binom{5}{2}=10\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose two elements is \(\binom{5}{2}=10\). Order is not counted in subsets.
Step 3
Exam Tip
दो अवयव चुनने की संख्या \(\binom{5}{2}=10\) है। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A=\{a,b,c\}\) है तो (A) के कुल उपसमुच्चयों की संख्या क्या होगी?
If \(A=\{a,b,c\}\), what is the total number of subsets of (A)?
#sets
#subsets
#power_set
A (3)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
A set with three elements has \(2^3=8\) subsets. Count the empty set and the set itself among all subsets.
Step 2
Why this answer is correct
The correct answer is C. (8). A set with three elements has \(2^3=8\) subsets. Count the empty set and the set itself among all subsets.
Step 3
Exam Tip
तीन अवयवों वाले समुच्चय के उपसमुच्चय \(2^3=8\) होते हैं। कुल उपसमुच्चय में रिक्त समुच्चय और वही समुच्चय भी गिनें।
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यदि \(A=\{1,2,3,4,5,6,7\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (2) और (5) अवश्य हों लेकिन (7) न हो?
If \(A=\{1,2,3,4,5,6,7\}\) then how many subsets contain (2) and (5) necessarily but do not contain (7)?
#sets
#subsets
#conditional-counting
A (8)
B (12)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
(2) and (5) are fixed and (7) is excluded, while the remaining (4) elements are free. Hence the number is \(2^4=16\).
Step 2
Why this answer is correct
The correct answer is C. (16). (2) and (5) are fixed and (7) is excluded, while the remaining (4) elements are free. Hence the number is \(2^4=16\).
Step 3
Exam Tip
(2) और (5) निश्चित हैं तथा (7) हटेगा, बाकी (4) अवयव स्वतंत्र हैं। इसलिए संख्या \(2^4=16\) है।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के (2) अवयवों वाले ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (5) न हो?
If \(A=\{1,2,3,4,5\}\) then how many two element subsets of (A) do not contain (5)?
#sets
#k-element-subsets
#excluded-element
A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
Exclude (5) and choose (2) elements from (1,2,3,4). The count is \(\binom{4}{2}=6\).
Step 2
Why this answer is correct
The correct answer is B. (6). Exclude (5) and choose (2) elements from (1,2,3,4). The count is \(\binom{4}{2}=6\).
Step 3
Exam Tip
(5) को हटाकर (1,2,3,4) में से (2) अवयव चुनने हैं। संख्या \(\binom{4}{2}=6\) है।
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यदि \(A=\{1,2,3,4,5,6\}\) है तो (A) के (3) अवयवों वाले ऐसे उपसमुच्चयों की संख्या क्या है जिनमें (1) अवश्य हो?
If \(A=\{1,2,3,4,5,6\}\) then how many three element subsets of (A) must contain (1)?
#sets
#k-element-subsets
#conditional-counting
A (5)
B (10)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed, so choose (2) elements from the remaining (5). The count is \(\binom{5}{2}=10\).
Step 2
Why this answer is correct
The correct answer is B. (10). (1) is fixed, so choose (2) elements from the remaining (5). The count is \(\binom{5}{2}=10\).
Step 3
Exam Tip
(1) निश्चित है, इसलिए बाकी (5) अवयवों में से (2) चुनने हैं। संख्या \(\binom{5}{2}=10\) है।
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यदि \(A=\{1,3,5,7\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनका योग विषम है?
If \(A=\{1,3,5,7\}\) then how many subsets have an odd sum?
#sets
#subsets
#odd-sum
A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
All elements are odd, so an odd sum needs an odd number of elements in the subset. \(\binom{4}{1}+\binom{4}{3}=4+4=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). All elements are odd, so an odd sum needs an odd number of elements in the subset. \(\binom{4}{1}+\binom{4}{3}=4+4=8\).
Step 3
Exam Tip
सभी अवयव विषम हैं, इसलिए विषम योग के लिए उपसमुच्चय में विषम संख्या में अवयव होने चाहिए। \(\binom{4}{1}+\binom{4}{3}=4+4=8\) है।
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यदि \(A=\{1,2,3,4\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनका योग सम है?
If \(A=\{1,2,3,4\}\) then how many subsets have an even sum?
#sets
#subsets
#parity
A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
There are two odd and two even elements, and an even sum needs an even number of odd elements. This gives \(2\times4=8\) subsets.
Step 2
Why this answer is correct
The correct answer is C. (8). There are two odd and two even elements, and an even sum needs an even number of odd elements. This gives \(2\times4=8\) subsets.
Step 3
Exam Tip
दो विषम और दो सम अवयव हैं, और सम योग के लिए विषम अवयवों की संख्या सम होनी चाहिए। कुल \(2\times4=8\) उपसमुच्चय मिलते हैं।
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यदि \(A=\{a,b,c,d,e\}\) है तो सभी उपसमुच्चयों में अवयव (c) कितनी बार आएगा?
If \(A=\{a,b,c,d,e\}\) then how many times will element (c) appear in all subsets?
#sets
#subsets
#element-frequency
A (8)
B (16)
C (24)
D (32)
Explanation opens after your attempt
Step 1
Concept
In a five element set, any fixed element appears in \(2^4=16\) subsets. Fix one element and let the other four vary freely.
Step 2
Why this answer is correct
The correct answer is B. (16). In a five element set, any fixed element appears in \(2^4=16\) subsets. Fix one element and let the other four vary freely.
Step 3
Exam Tip
पांच अवयवों वाले समुच्चय में कोई निश्चित अवयव \(2^{4}=16\) उपसमुच्चयों में आता है। एक अवयव निश्चित रखकर बाकी चार को स्वतंत्र मानें।
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यदि \(A=\{0,1,2,3\}\) है तो (A) के सभी उपसमुच्चयों में कुल कितनी बार अवयव (0) आएगा?
If \(A=\{0,1,2,3\}\) then in all subsets of (A), how many times will the element (0) appear in total?
#sets
#subsets
#element-frequency
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
In a (4)-element set, each element appears in \(2^{3}=8\) subsets. Use symmetry.
Step 2
Why this answer is correct
The correct answer is C. (8). In a (4)-element set, each element appears in \(2^{3}=8\) subsets. Use symmetry.
Step 3
Exam Tip
किसी (4) अवयव वाले समुच्चय में हर अवयव \(2^{3}=8\) उपसमुच्चयों में आता है। सममिति का उपयोग करें।
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यदि \(A=\{a,b,c,d\}\) है तो (A) के कम से कम तीन अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{a,b,c,d\}\) then what is the number of subsets of (A) having at least three elements?
#sets
#subsets
#at-least
A (4)
B (5)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
At least three elements means three or four elements. The count is \(\binom{4}{3}+\binom{4}{4}=4+1=5\).
Step 2
Why this answer is correct
The correct answer is B. (5). At least three elements means three or four elements. The count is \(\binom{4}{3}+\binom{4}{4}=4+1=5\).
Step 3
Exam Tip
कम से कम तीन अवयव का अर्थ तीन या चार अवयव हैं। संख्या \(\binom{4}{3}+\binom{4}{4}=4+1=5\) है।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के अधिकतम दो अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2,3,4,5\}\) then what is the number of subsets of (A) having at most two elements?
#sets
#subsets
#at-most
A (10)
B (15)
C (16)
D (20)
Explanation opens after your attempt
Step 1
Concept
At most two elements means (0,1,2) elements. The count is \(\binom{5}{0}+\binom{5}{1}+\binom{5}{2}=1+5+10=16\).
Step 2
Why this answer is correct
The correct answer is C. (16). At most two elements means (0,1,2) elements. The count is \(\binom{5}{0}+\binom{5}{1}+\binom{5}{2}=1+5+10=16\).
Step 3
Exam Tip
अधिकतम दो अवयव का अर्थ (0,1,2) अवयव हैं। संख्या \(\binom{5}{0}+\binom{5}{1}+\binom{5}{2}=1+5+10=16\) है।
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यदि \(A={x:x\in \mathbb{N}, x\leq 4}\) है तो (A) के उन उपसमुच्चयों की संख्या कितनी है जिनमें (4) हो?
If \(A={x:x\in \mathbb{N}, x\leq 4}\) then how many subsets of (A) contain (4)?
#sets
#conditional-subsets
#natural-numbers
A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
\(A=\{1,2,3,4\}\), and after fixing (4), the remaining (3) elements are free. Hence \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). \(A=\{1,2,3,4\}\), and after fixing (4), the remaining (3) elements are free. Hence \(2^3=8\).
Step 3
Exam Tip
\(A=\{1,2,3,4\}\) है और (4) निश्चित रखने पर बाकी (3) अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) है।
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यदि \(A=\{1,2,3,4\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (1) और (2) साथ साथ हों या दोनों न हों?
If \(A=\{1,2,3,4\}\) then how many subsets contain (1) and (2) together or contain neither?
#sets
#conditional-subsets
#counting
A (4)
B (6)
C (8)
D (10)
Explanation opens after your attempt
Step 1
Concept
When both (1,2) are included there are \(2^2=4\) choices, and when neither is included there are \(2^2=4\) choices. Total is (8).
Step 2
Why this answer is correct
The correct answer is C. (8). When both (1,2) are included there are \(2^2=4\) choices, and when neither is included there are \(2^2=4\) choices. Total is (8).
Step 3
Exam Tip
(1,2) दोनों होने पर \(2^2=4\) और दोनों न होने पर \(2^2=4\) विकल्प हैं। कुल (8) उपसमुच्चय बनते हैं।
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यदि \(A=\{p,q,r,s\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (p) या (q) में से कम से कम एक हो?
If \(A=\{p,q,r,s\}\) then how many subsets contain at least one of (p) or (q)?
#sets
#conditional-subsets
#at-least-one
A (8)
B (10)
C (12)
D (14)
Explanation opens after your attempt
Step 1
Concept
Total subsets are (16), and those containing neither (p) nor (q) are \(2^2=4\). Hence the answer is (16-4=12).
Step 2
Why this answer is correct
The correct answer is C. (12). Total subsets are (16), and those containing neither (p) nor (q) are \(2^2=4\). Hence the answer is (16-4=12).
Step 3
Exam Tip
कुल उपसमुच्चय (16) हैं और जिनमें (p,q) दोनों नहीं हैं वे \(2^2=4\) हैं। अतः उत्तर (16-4=12) है।
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यदि \(A=\{1,2,3,4,5,6\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (1) न हो और (6) अवश्य हो?
If \(A=\{1,2,3,4,5,6\}\) then how many subsets do not contain (1) but must contain (6)?
#sets
#conditional-subsets
#counting
A (8)
B (16)
C (32)
D (64)
Explanation opens after your attempt
Step 1
Concept
(1) is excluded and (6) is fixed, while (2,3,4,5) are free. So \(2^4=16\) subsets are formed.
Step 2
Why this answer is correct
The correct answer is B. (16). (1) is excluded and (6) is fixed, while (2,3,4,5) are free. So \(2^4=16\) subsets are formed.
Step 3
Exam Tip
(1) हटेगा और (6) निश्चित होगा, बाकी (2,3,4,5) स्वतंत्र हैं। इसलिए \(2^4=16\) उपसमुच्चय बनते हैं।
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यदि \(A=\{a,b,c,d,e\}\) है तो (A) के ठीक तीन अवयवों वाले उपसमुच्चयों की संख्या क्या है?
If \(A=\{a,b,c,d,e\}\) then what is the number of subsets of (A) having exactly three elements?
#sets
#k-element-subsets
#combinations
A (5)
B (10)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose three elements is \(\binom{5}{3}=10\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose three elements is \(\binom{5}{3}=10\). Order is not counted in subsets.
Step 3
Exam Tip
तीन अवयव चुनने की संख्या \(\binom{5}{3}=10\) है। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A=\{2,3,5,7\}\) है तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (2) हो पर (7) न हो?
If \(A=\{2,3,5,7\}\) then how many subsets contain (2) but do not contain (7)?
#sets
#conditional-subsets
#counting
A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(2) is fixed and (7) is excluded, while (3,5) are free. Hence the number is \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is B. (4). (2) is fixed and (7) is excluded, while (3,5) are free. Hence the number is \(2^2=4\).
Step 3
Exam Tip
(2) निश्चित है और (7) हटेगा, बाकी (3,5) स्वतंत्र हैं। इसलिए संख्या \(2^2=4\) है।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के दो अवयवों वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3,4,5\}\) then how many two element subsets does (A) have?
#sets
#two-element-subsets
#combinations
A (5)
B (10)
C (15)
D (20)
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose two elements is \(\binom{5}{2}=10\). Order is not counted in subsets.
Step 2
Why this answer is correct
The correct answer is B. (10). The number of ways to choose two elements is \(\binom{5}{2}=10\). Order is not counted in subsets.
Step 3
Exam Tip
दो अवयव चुनने की संख्या \(\binom{5}{2}=10\) होती है। उपसमुच्चय में क्रम नहीं गिना जाता।
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यदि \(A=\{a,b,c,d\}\) है तो कितने उपसमुच्चयों में (a) नहीं होगा?
If \(A=\{a,b,c,d\}\) then how many subsets will not contain (a)?
#sets
#subsets
#counting
A (4)
B (8)
C (12)
D (16)
Explanation opens after your attempt
Step 1
Concept
After excluding (a), subsets are formed from (b,c,d). So the number is \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is B. (8). After excluding (a), subsets are formed from (b,c,d). So the number is \(2^3=8\).
Step 3
Exam Tip
(a) को हटाने के बाद (b,c,d) से उपसमुच्चय बनेंगे। अतः संख्या \(2^3=8\) है।
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यदि \(A=\{1,2,3,4\}\) है तो कितने उपसमुच्चयों में अवयव (1) अवश्य होगा?
If \(A=\{1,2,3,4\}\) then how many subsets must contain the element (1)?
#sets
#subsets
#counting-with-condition
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
Fix (1) and choose freely from the remaining (3) elements. Hence the number is \(2^3=8\).
Step 2
Why this answer is correct
The correct answer is C. (8). Fix (1) and choose freely from the remaining (3) elements. Hence the number is \(2^3=8\).
Step 3
Exam Tip
(1) को निश्चित रखकर बाकी (3) अवयव स्वतंत्र रूप से चुने जा सकते हैं। इसलिए संख्या \(2^3=8\) है।
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कौन सा विकल्प ({1,2}) के सभी उपसमुच्चयों को सही दिखाता है?
Which option correctly shows all subsets of ({1,2})?
#sets
#power-set
#listing-subsets
A \(\varnothing,{1},{2},{1,2}\)
B ({1},{2})
C \(\varnothing,{1,2}\)
D ({1},{2},{3})
Explanation opens after your attempt
Correct Answer
A. \(\varnothing,{1},{2},{1,2}\)
Step 1
Concept
A two element set has (4) subsets. Include singletons and the whole set.
Step 2
Why this answer is correct
The correct answer is A. \(\varnothing,{1},{2},{1,2}\). A two element set has (4) subsets. Include singletons and the whole set.
Step 3
Exam Tip
दो अवयवों वाले समुच्चय के (4) उपसमुच्चय होते हैं। एकल अवयव और पूरा समुच्चय दोनों शामिल करें।
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यदि \(A=\{1,2\}\) है तो (A) के कुल उपसमुच्चयों की संख्या क्या है?
If \(A=\{1,2\}\) then what is the total number of subsets of (A)?
#sets
#number-of-subsets
#power-set
A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
If a set has (n) elements then it has \(2^n\) subsets. Here \(2^2=4\).
Step 2
Why this answer is correct
The correct answer is C. (4). If a set has (n) elements then it has \(2^n\) subsets. Here \(2^2=4\).
Step 3
Exam Tip
यदि समुच्चय में (n) अवयव हों तो उपसमुच्चय \(2^n\) होते हैं। यहां \(2^2=4\) है।
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यदि किसी समुच्चय में (4) अवयव हैं, तो उसके कुल उपसमुच्चयों की संख्या कितनी होगी?
If a set has (4) elements, how many subsets will it have?
#power-set
#subsets
#counting
#class11
A (8)
B (12)
C (16)
D (4)
Explanation opens after your attempt
Step 1
Concept
A set with (n) elements has \(2^n\) subsets.
Step 2
Why this answer is correct
Here (n=4), so \(2^4=16\).
Step 3
Exam Tip
While counting, include the empty set and the whole set too. चरण 1: (n) अवयवों वाले समुच्चय के उपसमुच्चयों की संख्या \(2^n\) होती है। चरण 2: यहाँ (n=4), इसलिए \(2^4=16\)। चरण 3: संख्या गिनते समय रिक्त समुच्चय और पूरा समुच्चय भी शामिल करें।
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यदि \(A=\{p,q,r\}\), तो (A) के एक अवयव वाले उपसमुच्चय कितने हैं?
If \(A=\{p,q,r\}\), how many one-element subsets does (A) have?
#sets
#singleton subsets
#counting
A (1)
B (2)
C (3)
D (6)
Explanation opens after your attempt
Step 1
Concept
One-element subsets are formed by taking one element at a time.
Step 2
Why this answer is correct
({p},{q},{r}) are the three such subsets.
Step 3
Exam Tip
A set with (n) elements has (n) one-element subsets. चरण 1: एक अवयव वाले उपसमुच्चय एक-एक अवयव से बनते हैं। चरण 2: ({p},{q},{r}) तीन ऐसे उपसमुच्चय हैं। चरण 3: किसी समुच्चय में (n) अवयव हों तो एक अवयव वाले उपसमुच्चय (n) होते हैं।
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समुच्चय \(A=\{1,2,3\}\) के दो अवयवों वाले उपसमुच्चयों की संख्या कितनी है?
How many two-element subsets does \(A=\{1,2,3\}\) have?
#sets
#two element subsets
#counting
A (2)
B (3)
C (4)
D (6)
Explanation opens after your attempt
Step 1
Concept
We need subsets containing exactly two elements.
Step 2
Why this answer is correct
They are ({1,2},{1,3},{2,3}), so there are (3).
Step 3
Exam Tip
Order does not matter in a subset. चरण 1: दो अवयवों वाले उपसमुच्चय चुनने हैं। चरण 2: वे ({1,2},{1,3},{2,3}) हैं, कुल (3)। चरण 3: उपसमुच्चय में क्रम का महत्व नहीं होता।
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यदि \(A=\{a,b,c,d\}\), तो (A) के उचित उपसमुच्चयों की संख्या कितनी है?
If \(A=\{a,b,c,d\}\), how many proper subsets does (A) have?
#sets
#proper subsets
#counting
A (15)
B (16)
C (8)
D (4)
Explanation opens after your attempt
Step 1
Concept
The total number of subsets is \(2^4=16\).
Step 2
Why this answer is correct
A proper subset does not include the set itself, so (16-1=15).
Step 3
Exam Tip
When proper subsets are asked, subtract the original set. चरण 1: कुल उपसमुच्चय \(2^4=16\) होंगे। चरण 2: उचित उपसमुच्चय में पूरा समुच्चय स्वयं शामिल नहीं होता, इसलिए (16-1=15)। चरण 3: उचित उपसमुच्चय पूछे जाने पर पूरे समुच्चय को घटाएँ।
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दो अवयवों वाले समुच्चय के उचित उपसमुच्चयों की संख्या कितनी होती है?
How many proper subsets does a set with two elements have?
#proper-subsets
#counting
#sets
A (1)
B (2)
C (3)
D (4)
Explanation opens after your attempt
Step 1
Concept
A set with (2) elements has \(2^2=4\) subsets.
Step 2
Why this answer is correct
For proper subsets, the set itself is not counted, so (4-1=3).
Step 3
Exam Tip
Exam tip: when proper subsets are asked, subtract the original set. चरण 1: (2) अवयवों वाले समुच्चय के कुल उपसमुच्चय \(2^2=4\) होते हैं। चरण 2: उचित उपसमुच्चयों में पूरा समुच्चय नहीं गिना जाता, इसलिए (4-1=3)। चरण 3: परीक्षा संकेत: उचित उपसमुच्चय पूछे तो पूरे समुच्चय को हटाएं।
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यदि किसी समुच्चय में (4) अवयव हैं, तो उसके कुल उपसमुच्चय कितने होंगे?
If a set has (4) elements, how many subsets does it have?
#number-of-subsets
#finite-set
#sets
A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
A set with (n) elements has \(2^n\) subsets.
Step 2
Why this answer is correct
Here (n=4), so \(2^4=16\).
Step 3
Exam Tip
Exam tip: for each element, think of two choices: include it or exclude it. चरण 1: (n) अवयवों वाले समुच्चय के उपसमुच्चय \(2^n\) होते हैं। चरण 2: यहाँ (n=4), इसलिए \(2^4=16\)। चरण 3: परीक्षा संकेत: हर अवयव के लिए दो स्थितियां सोचें, लेना या न लेना।
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समुच्चय \(A=\{a,b,c\}\) के कुल कितने उपसमुच्चय होंगे?
How many subsets will the set \(A=\{a,b,c\}\) have?
#sets
#number-of-subsets
#power-set
#easy
A (3)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
If a set has (n) elements, the number of subsets is \(2^n\).
Step 2
Why this answer is correct
Here (n=3), so \(2^3=8\).
Step 3
Exam Tip
Exam tip: include the empty set and the set itself while counting. चरण 1: यदि किसी समुच्चय में (n) अवयव हों, तो उपसमुच्चयों की संख्या \(2^n\) होती है। चरण 2: यहाँ (n=3), इसलिए \(2^3=8\)। चरण 3: परीक्षा संकेत: खाली समुच्चय और पूरा समुच्चय भी गिनना न भूलें।
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समुच्चय \(A=\{3\}\) के उपसमुच्चय कौन-से हैं?
What are the subsets of \(A=\{3\}\)?
#singleton-set
#subsets
#counting
A \(\varnothing,{3}\)
B ({0},{3})
C ({3},{3,3})
D \(\varnothing,{0},{3}\)
Explanation opens after your attempt
Correct Answer
A. \(\varnothing,{3}\)
Step 1
Concept
A singleton set has two subsets.
Step 2
Why this answer is correct
They are the empty set and the singleton set itself.
Step 3
Exam Tip
Repeating an element does not create a new subset. चरण 1: एक-सदस्यीय समुच्चय के दो उपसमुच्चय होते हैं। चरण 2: वे हैं रिक्त समुच्चय और वही एक-सदस्यीय समुच्चय। चरण 3: किसी सदस्य को दो बार लिखने से नया उपसमुच्चय नहीं बनता।
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समुच्चय \(A=\{1,3,5\}\) के कुल उपसमुच्चयों की संख्या कितनी है?
How many subsets does the set \(A=\{1,3,5\}\) have?
#number-of-subsets
#power-set
#sets
A (3)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
If a finite set has (n) elements, the number of subsets is \(2^n\).
Step 2
Why this answer is correct
Here (n=3), so \(2^3=8\).
Step 3
Exam Tip
Count the elements first and then apply \(2^n\). चरण 1: यदि किसी सीमित समुच्चय में (n) सदस्य हों, तो उपसमुच्चयों की संख्या \(2^n\) होती है। चरण 2: यहाँ (n=3), इसलिए \(2^3=8\)। चरण 3: सदस्य गिनने के बाद सीधे \(2^n\) लगाएँ।
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यदि \(A=\{1,2,3\}\) है, तो (A) के कुल उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3\}\), how many total subsets does (A) have?
#sets
#subsets
#counting
#class11
A (3)
B (6)
C (8)
D (9)
Explanation opens after your attempt
Step 1
Concept
The set (A) has (3) elements.
Step 2
Why this answer is correct
The total number of subsets is \(2^3=8\).
Step 3
Exam Tip
In counting questions, first identify the number of elements. चरण 1: समुच्चय (A) में (3) अवयव हैं। चरण 2: कुल उपसमुच्चय \(2^3=8\) होंगे। चरण 3: गिनती वाले प्रश्नों में पहले अवयवों की संख्या पहचानें।
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यदि \(A=\{1,2,3,4,5,6\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जिनकी cardinality (2) से विभाज्य है?
If \(A=\{1,2,3,4,5,6\}\), how many subsets in (\mathcal{P}(A)) have cardinality divisible by (2)?
#sets
#even-subsets
#power-set
A (16)
B (32)
C (48)
D (64)
Explanation opens after your attempt
Step 1
Concept
Cardinality divisible by (2) means even cardinality. A (6)-element set has \(2^{6-1}=32\) even subsets.
Step 2
Why this answer is correct
The correct answer is B. (32). Cardinality divisible by (2) means even cardinality. A (6)-element set has \(2^{6-1}=32\) even subsets.
Step 3
Exam Tip
Cardinality (2) से विभाज्य होने का अर्थ even cardinality है। (6)-element set के even subsets \(2^{6-1}=32\) होते हैं।
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यदि \(A=\{1,2,3,4,5\}\) है, तो (\mathcal{P}(A)) में ऐसे subsets कितने हैं जो ({1,2}) से disjoint हैं?
If \(A=\{1,2,3,4,5\}\), how many subsets in (\mathcal{P}(A)) are disjoint from ({1,2})?
#sets
#disjoint-subsets
#power-set
A (4)
B (8)
C (16)
D (32)
Explanation opens after your attempt
Step 1
Concept
Disjoint subsets are formed only from ({3,4,5}), so there are \(2^3=8\). In exams, remove forbidden elements for a disjoint condition.
Step 2
Why this answer is correct
The correct answer is B. (8). Disjoint subsets are formed only from ({3,4,5}), so there are \(2^3=8\). In exams, remove forbidden elements for a disjoint condition.
Step 3
Exam Tip
Disjoint subsets केवल ({3,4,5}) से बनेंगे, इसलिए \(2^3=8\) हैं। परीक्षा में disjoint condition के लिए forbidden elements हटा दें।
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यदि (|A|=7) है, तो (\mathcal{P}(A)) में even cardinality वाले subsets की संख्या कितनी है?
If (|A|=7), how many subsets in (\mathcal{P}(A)) have even cardinality?
#sets
#even-subsets
#power-set
A (32)
B (64)
C (96)
D (128)
Explanation opens after your attempt
Step 1
Concept
When \(|A|=n\geq1\), the number of even cardinality subsets is \(2^{n-1}\). Here it is \(2^6=64\).
Step 2
Why this answer is correct
The correct answer is B. (64). When \(|A|=n\geq1\), the number of even cardinality subsets is \(2^{n-1}\). Here it is \(2^6=64\).
Step 3
Exam Tip
जब \(|A|=n\geq1\), even cardinality subsets की संख्या \(2^{n-1}\) होती है। यहां \(2^6=64\) है।
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यदि (|A|=8) है, तो (\mathcal{P}(A)) में even cardinality वाले subsets कितने होंगे?
If (|A|=8), how many subsets in (\mathcal{P}(A)) have even cardinality?
#sets
#even-subsets
#cardinality
A (64)
B (128)
C (256)
D (512)
Explanation opens after your attempt
Step 1
Concept
The number of even cardinality subsets is \(2^{8-1}=128\). In exams, remember that for \(n\geq1\), even and odd subsets are equal.
Step 2
Why this answer is correct
The correct answer is B. (128). The number of even cardinality subsets is \(2^{8-1}=128\). In exams, remember that for \(n\geq1\), even and odd subsets are equal.
Step 3
Exam Tip
Even cardinality subsets की संख्या \(2^{8-1}=128\) है। परीक्षा में \(n\geq1\) होने पर even और odd subsets बराबर याद रखें।
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यदि (A) में (7) तत्व हैं, तो (\mathcal{P}(A)) में विषम संख्या के तत्वों वाले subsets कितने हैं?
If (A) has (7) elements, how many subsets in (\mathcal{P}(A)) have an odd number of elements?
#sets
#odd-subsets
#power-set
A (32)
B (64)
C (96)
D (128)
Explanation opens after your attempt
Step 1
Concept
For an (n)-element set, the number of odd subsets is \(2^{n-1}\), so \(2^6=64\). In exams, even and odd subsets are equal in number.
Step 2
Why this answer is correct
The correct answer is B. (64). For an (n)-element set, the number of odd subsets is \(2^{n-1}\), so \(2^6=64\). In exams, even and odd subsets are equal in number.
Step 3
Exam Tip
किसी (n)-तत्वीय समुच्चय में odd subsets की संख्या \(2^{n-1}\) होती है, अतः \(2^6=64\)। परीक्षा में even और odd subsets बराबर होते हैं।
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यदि किसी समुच्चय (B) के कुल उपसमुच्चय (1024) हैं, तो उसके उचित उपसमुच्चयों की संख्या कितनी होगी?
If a set (B) has (1024) total subsets, how many proper subsets does it have?
#sets
#proper subset
#power set
#class 11
A (1023)
B (1024)
C (512)
D (10)
Explanation opens after your attempt
Step 1
Concept
A proper subset does not include the whole set itself. Therefore the number is (1024-1=1023).
Step 2
Why this answer is correct
The correct answer is A. (1023). A proper subset does not include the whole set itself. Therefore the number is (1024-1=1023).
Step 3
Exam Tip
उचित उपसमुच्चय में पूरा समुच्चय स्वयं शामिल नहीं होता। इसलिए संख्या (1024-1=1023) है।
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यदि (A) में (3) तत्व हैं, तो (A) के अरिक्त उचित उपसमुच्चयों की संख्या कितनी होगी?
If (A) has (3) elements, how many non-empty proper subsets of (A) are there?
#sets
#proper subsets
#non empty subsets
#class 11
A (3)
B (6)
C (7)
D (8)
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^3=8\). Removing \(\varnothing\) and (A) gives (6) non-empty proper subsets.
Step 2
Why this answer is correct
The correct answer is B. (6). Total subsets are \(2^3=8\). Removing \(\varnothing\) and (A) gives (6) non-empty proper subsets.
Step 3
Exam Tip
कुल उपसमुच्चय \(2^3=8\) हैं। अरिक्त उचित उपसमुच्चयों के लिए \(\varnothing\) और (A) हटाने पर (6) बचते हैं।
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यदि \(A=\{2,4,6,8\}\) है तो कुल उपसमुच्चयों की संख्या और दो तत्व वाले उपसमुच्चयों की संख्या क्रमशः क्या है?
If \(A=\{2,4,6,8\}\), what are the total number of subsets and the number of two element subsets respectively?
#sets
#power_set
#mixed_count
A (8) और (4) / (8) and (4)
B (16) और (6) / (16) and (6)
C (16) और (8) / (16) and (8)
D (4) और (16) / (4) and (16)
Explanation opens after your attempt
Correct Answer
B. (16) और (6) / (16) and (6)
Step 1
Concept
A set with four elements has \(2^4=16\) total subsets. The number of two element subsets is (6).
Step 2
Why this answer is correct
The correct answer is B. (16) और (6) / (16) and (6). A set with four elements has \(2^4=16\) total subsets. The number of two element subsets is (6).
Step 3
Exam Tip
चार तत्वों वाले समुच्चय के कुल \(2^4=16\) उपसमुच्चय होते हैं। दो तत्व वाले उपसमुच्चय (6) होते हैं।
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यदि \(A=\{1,3,5,7\}\) है तो (\mathcal{P}(A)) में कुल उपसमुच्चयों की संख्या और एक तत्व वाले उपसमुच्चयों की संख्या क्रमशः क्या है?
If \(A=\{1,3,5,7\}\), what are the total number of subsets and the number of one element subsets in (\mathcal{P}(A)) respectively?
#sets
#power_set
#mixed_count
A (8) और (4) / (8) and (4)
B (16) और (4) / (16) and (4)
C (16) और (8) / (16) and (8)
D (4) और (16) / (4) and (16)
Explanation opens after your attempt
Correct Answer
B. (16) और (4) / (16) and (4)
Step 1
Concept
A set with four elements has \(2^4=16\) total subsets. The number of one element subsets is (4).
Step 2
Why this answer is correct
The correct answer is B. (16) और (4) / (16) and (4). A set with four elements has \(2^4=16\) total subsets. The number of one element subsets is (4).
Step 3
Exam Tip
चार तत्वों वाले समुच्चय के कुल \(2^4=16\) उपसमुच्चय होते हैं। एक तत्व वाले उपसमुच्चय (4) होते हैं।
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यदि \(A={x:x\in\mathbb{Z},x^2\le4}\), तो (A) के उचित उपसमुच्चयों की संख्या कितनी है?
If \(A={x:x\in\mathbb{Z},x^2\le4}\), how many proper subsets does (A) have?
#proper-subset
#counting
#integers
A (15)
B (31)
C (32)
D (63)
Explanation opens after your attempt
Step 1
Concept
\(A=\{-2,-1,0,1,2\}\) has (5) elements. Proper subsets are \(2^5-1=31\).
Step 2
Why this answer is correct
The correct answer is B. (31). \(A=\{-2,-1,0,1,2\}\) has (5) elements. Proper subsets are \(2^5-1=31\).
Step 3
Exam Tip
\(A=\{-2,-1,0,1,2\}\) में (5) सदस्य हैं। उचित उपसमुच्चय \(2^5-1=31\) होंगे।
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यदि \(A=\{a,b,c,d,e\}\), तो (a) को शामिल न करने वाले उचित उपसमुच्चयों की संख्या कितनी है?
If \(A=\{a,b,c,d,e\}\), how many proper subsets do not contain (a)?
#proper-subset
#counting
#excluded-element
A (8)
B (15)
C (16)
D (31)
Explanation opens after your attempt
Step 1
Concept
After excluding (a), (4) elements remain, giving \(2^4=16\) subsets. All of them are proper subsets of (A).
Step 2
Why this answer is correct
The correct answer is C. (16). After excluding (a), (4) elements remain, giving \(2^4=16\) subsets. All of them are proper subsets of (A).
Step 3
Exam Tip
(a) को हटाकर (4) सदस्य बचते हैं, जिनके \(2^4=16\) उपसमुच्चय हैं। ये सभी (A) के उचित उपसमुच्चय हैं।
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यदि \(A={\emptyset,{\emptyset}}\) है, तो (A) के उपसमुच्चयों की संख्या कितनी है?
If \(A={\emptyset,{\emptyset}}\), how many subsets does (A) have?
#empty-set
#nested-set
#counting
A (2)
B (3)
C (4)
D (5)
Explanation opens after your attempt
Step 1
Concept
The set (A) has two distinct elements: \(\emptyset\) and \({\emptyset}\). Hence it has \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is C. (4). The set (A) has two distinct elements: \(\emptyset\) and \({\emptyset}\). Hence it has \(2^2=4\) subsets.
Step 3
Exam Tip
(A) में दो अलग सदस्य हैं: \(\emptyset\) और \({\emptyset}\)। इसलिए उपसमुच्चय \(2^2=4\) हैं।
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यदि (A) में (3) सदस्य हैं, तो (\mathcal{P}(A)) के उचित उपसमुच्चयों की संख्या कितनी है?
If (A) has (3) elements, how many proper subsets does (\mathcal{P}(A)) have?
#power-set
#counting
#proper-subset
A (127)
B (255)
C (256)
D (511)
Explanation opens after your attempt
Step 1
Concept
(\mathcal{P}(A)) has \(2^3=8\) elements. Its proper subsets are \(2^8-1=255\).
Step 2
Why this answer is correct
The correct answer is B. (255). (\mathcal{P}(A)) has \(2^3=8\) elements. Its proper subsets are \(2^8-1=255\).
Step 3
Exam Tip
(\mathcal{P}(A)) में \(2^3=8\) सदस्य हैं। इसके उचित उपसमुच्चय \(2^8-1=255\) होंगे।
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यदि \(A=\{1,2,3,4,5,6\}\) है तो (A) के कितने उपसमुच्चय केवल सम संख्याओं से बन सकते हैं?
If \(A=\{1,2,3,4,5,6\}\) then how many subsets of (A) can be formed using only even numbers?
#sets
#subset counting
#even numbers
A (3)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
The even elements are (2,4,6), and all their subsets are \(2^3=8\). In exams "only" means all other elements are excluded.
Step 2
Why this answer is correct
The correct answer is C. (8). The even elements are (2,4,6), and all their subsets are \(2^3=8\). In exams "only" means all other elements are excluded.
Step 3
Exam Tip
सम तत्व (2,4,6) हैं और इनके सभी उपसमुच्चय \(2^3=8\) होंगे। परीक्षा में "केवल" का अर्थ है बाकी तत्वों को पूरी तरह हटाना।
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यदि \(A=\{1,2,3,4\}\) है तो कितने उपसमुच्चयों में ठीक एक विषम संख्या है?
If \(A=\{1,2,3,4\}\) then how many subsets contain exactly one odd number?
#sets
#subset counting
#odd numbers
A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
The odd numbers are (1,3); choose one of them and let even numbers (2,4) be free, so \({}^2C_1\times2^2=8\). In exams separate restricted and free elements.
Step 2
Why this answer is correct
The correct answer is D. (8). The odd numbers are (1,3); choose one of them and let even numbers (2,4) be free, so \({}^2C_1\times2^2=8\). In exams separate restricted and free elements.
Step 3
Exam Tip
विषम संख्याएं (1,3) हैं जिनमें से एक चुनें और सम संख्याएं (2,4) स्वतंत्र हैं इसलिए \({}^2C_1\times2^2=8\)। परीक्षा में शर्त वाले तत्व और स्वतंत्र तत्व अलग करें।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के कितने उपसमुच्चय (1) रखते हैं लेकिन (4) नहीं रखते?
If \(A=\{1,2,3,4\}\) then how many subsets of (A) contain (1) but not (4)?
#sets
#subset counting
#conditions
A (2)
B (4)
C (6)
D (8)
Explanation opens after your attempt
Step 1
Concept
(1) is fixed and (4) is removed, so (2,3) are free and form \(2^2=4\) subsets. In exams separate included and excluded conditions.
Step 2
Why this answer is correct
The correct answer is B. (4). (1) is fixed and (4) is removed, so (2,3) are free and form \(2^2=4\) subsets. In exams separate included and excluded conditions.
Step 3
Exam Tip
(1) निश्चित है और (4) हट गया है इसलिए (2,3) स्वतंत्र हैं और \(2^2=4\) उपसमुच्चय बनते हैं। परीक्षा में शामिल और निष्कासित शर्तें अलग करें।
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यदि \(A=\{1,2,3,4,5\}\) है तो (A) के कितने उपसमुच्चय (2) और (5) दोनों को रखते हैं?
If \(A=\{1,2,3,4,5\}\) then how many subsets of (A) contain both (2) and (5)?
#sets
#subset counting
#compulsory elements
A (4)
B (6)
C (8)
D (16)
Explanation opens after your attempt
Step 1
Concept
The elements (2) and (5) are fixed and the remaining (3) elements are free, so \(2^3=8\). In exams fix compulsory elements first.
Step 2
Why this answer is correct
The correct answer is C. (8). The elements (2) and (5) are fixed and the remaining (3) elements are free, so \(2^3=8\). In exams fix compulsory elements first.
Step 3
Exam Tip
(2) और (5) निश्चित हैं और बाकी (3) तत्व स्वतंत्र हैं इसलिए \(2^3=8\)। परीक्षा में अनिवार्य तत्वों को पहले निश्चित करें।
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यदि \(A=\{1,2,3\}\) है तो (A) के कम से कम दो तत्वों वाले उपसमुच्चयों की संख्या कितनी है?
If \(A=\{1,2,3\}\) then how many subsets of (A) have at least two elements?
#sets
#subset counting
#at least
A (3)
B (4)
C (5)
D (6)
Explanation opens after your attempt
Step 1
Concept
There are (3) subsets with two elements and (1) subset with three elements, so the total is (4). In exams add all larger sizes for "at least".
Step 2
Why this answer is correct
The correct answer is B. (4). There are (3) subsets with two elements and (1) subset with three elements, so the total is (4). In exams add all larger sizes for "at least".
Step 3
Exam Tip
दो तत्वों वाले (3) और तीन तत्वों वाला (1) उपसमुच्चय है इसलिए कुल (4) हैं। परीक्षा में "कम से कम" में सभी बड़े आकार जोड़ें।
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यदि \(A=\{1,2,3,4\}\) है तो (A) के कितने उपसमुच्चय (1) को नहीं रखते?
If \(A=\{1,2,3,4\}\) then how many subsets of (A) do not contain (1)?
#sets
#subset counting
#excluded element
A (4)
B (8)
C (12)
D (15)
Explanation opens after your attempt
Step 1
Concept
After removing (1), the remaining (3) elements form \(2^3=8\) subsets. In exams remove the forbidden element first.
Step 2
Why this answer is correct
The correct answer is B. (8). After removing (1), the remaining (3) elements form \(2^3=8\) subsets. In exams remove the forbidden element first.
Step 3
Exam Tip
(1) को हटाने पर बचे (3) तत्वों से \(2^3=8\) उपसमुच्चय बनते हैं। परीक्षा में निषिद्ध तत्व को पहले हटाएं।
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यदि \(A=\{p,q,r\}\) है तो (A) के कितने उपसमुच्चय (p) को अवश्य रखते हैं?
If \(A=\{p,q,r\}\) then how many subsets of (A) must contain (p)?
#sets
#subset counting
#fixed element
A (2)
B (3)
C (4)
D (8)
Explanation opens after your attempt
Step 1
Concept
The element (p) is fixed and the remaining two elements can be chosen in \(2^2\) ways. In exams fix the compulsory element and count the rest.
Step 2
Why this answer is correct
The correct answer is C. (4). The element (p) is fixed and the remaining two elements can be chosen in \(2^2\) ways. In exams fix the compulsory element and count the rest.
Step 3
Exam Tip
(p) निश्चित है और बाकी दो तत्वों को \(2^2\) तरीकों से चुना जा सकता है। परीक्षा में निश्चित तत्व अलग रखकर बाकी पर गिनती करें।
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यदि \(A=\{1,2,3\}\) और \(B=\{1,2,3,4,5\}\), तो (B) के ऐसे उपसमुच्चयों की संख्या कितनी है जो (A) को अवश्य समाहित करते हैं?
If \(A=\{1,2,3\}\) and \(B=\{1,2,3,4,5\}\), how many subsets of (B) must contain (A)?
#subset containing set
#counting
A 2
B 4
C 8
D 16
Explanation opens after your attempt
Step 1
Concept
1, 2 and 3 are fixed, while 4 and 5 are optional. Hence \(2^2=4\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 4. 1, 2 and 3 are fixed, while 4 and 5 are optional. Hence \(2^2=4\) subsets.
Step 3
Exam Tip
1, 2 और 3 निश्चित हैं, जबकि 4 और 5 वैकल्पिक हैं। इसलिए \(2^2=4\) उपसमुच्चय होंगे।
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यदि \(A=\{1,2,3,4,5\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 हो और 2 न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets contain 1 and do not contain 2?
#subset counting
#and condition
A 4
B 8
C 16
D 24
Explanation opens after your attempt
Step 1
Concept
1 is fixed and 2 is excluded, while the remaining 3 elements are optional. Thus \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. 1 is fixed and 2 is excluded, while the remaining 3 elements are optional. Thus \(2^3=8\) subsets.
Step 3
Exam Tip
1 निश्चित है और 2 नहीं लेना है, शेष 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय होंगे।
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यदि \(A=\{1,2,3,4\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 1 हो या 2 हो?
If \(A=\{1,2,3,4\}\), how many subsets contain 1 or 2?
#subset counting
#or condition
#hard
A 8
B 12
C 14
D 16
Explanation opens after your attempt
Step 1
Concept
There are 16 total subsets, and those containing neither 1 nor 2 are the \(2^2=4\) subsets of ({3,4}). Hence (16-4=12).
Step 2
Why this answer is correct
The correct answer is B. 12. There are 16 total subsets, and those containing neither 1 nor 2 are the \(2^2=4\) subsets of ({3,4}). Hence (16-4=12).
Step 3
Exam Tip
कुल 16 उपसमुच्चय हैं और जिनमें न 1 है न 2, वे ({3,4}) के \(2^2=4\) उपसमुच्चय हैं। इसलिए (16-4=12)।
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यदि \(A=\{1,2,3,4,5\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक 2 अवयव हों और 5 न हो?
If \(A=\{1,2,3,4,5\}\), how many subsets have exactly 2 elements and do not contain 5?
#exact subset
#counting
#combination
A 4
B 6
C 8
D 10
Explanation opens after your attempt
Step 1
Concept
Exclude 5 and choose 2 elements from ({1,2,3,4}). The number is \(\binom{4}{2}=6\).
Step 2
Why this answer is correct
The correct answer is B. 6. Exclude 5 and choose 2 elements from ({1,2,3,4}). The number is \(\binom{4}{2}=6\).
Step 3
Exam Tip
5 को छोड़कर ({1,2,3,4}) से 2 अवयव चुनने हैं। संख्या \(\binom{4}{2}=6\) है।
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यदि \(A=\{2,3,5,7,11\}\), तो (A) के ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 2 हो लेकिन 11 न हो?
If \(A=\{2,3,5,7,11\}\), how many subsets contain 2 but not 11?
#subset counting
#conditions
A 4
B 8
C 16
D 32
Explanation opens after your attempt
Step 1
Concept
2 is fixed and 11 is excluded, while the remaining 3 elements are optional. Hence \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. 2 is fixed and 11 is excluded, while the remaining 3 elements are optional. Hence \(2^3=8\) subsets.
Step 3
Exam Tip
2 निश्चित है और 11 निषिद्ध है, शेष 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय हैं।
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यदि \(A=\{1,2,3,4\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें ठीक 3 अवयव हों?
If \(A=\{1,2,3,4\}\), how many subsets have exactly 3 elements?
#exact size subset
#combination
A 3
B 4
C 6
D 8
Explanation opens after your attempt
Step 1
Concept
The number of ways to choose exactly 3 elements is \(\binom{4}{3}=4\). Changing order does not create a new subset.
Step 2
Why this answer is correct
The correct answer is B. 4. The number of ways to choose exactly 3 elements is \(\binom{4}{3}=4\). Changing order does not create a new subset.
Step 3
Exam Tip
ठीक 3 अवयव चुनने के तरीके \(\binom{4}{3}=4\) हैं। क्रम बदलने से नया उपसमुच्चय नहीं बनता।
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यदि \(A=\{a,b,c,d,e\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें (a) और (b) दोनों हों?
If \(A=\{a,b,c,d,e\}\), how many subsets contain both (a) and (b)?
#subset counting
#fixed elements
A 4
B 8
C 16
D 32
Explanation opens after your attempt
Step 1
Concept
(a) and (b) are fixed, while the other 3 elements are optional. Hence \(2^3=8\) subsets.
Step 2
Why this answer is correct
The correct answer is B. 8. (a) and (b) are fixed, while the other 3 elements are optional. Hence \(2^3=8\) subsets.
Step 3
Exam Tip
(a) और (b) निश्चित हैं, बाकी 3 अवयव स्वतंत्र हैं। इसलिए \(2^3=8\) उपसमुच्चय होंगे।
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