यदि \(A=\{p,q,r,s\}\), तो (\mathcal{P}(A)) में (p) नहीं रखने वाले कितने तत्व होंगे?
If \(A=\{p,q,r,s\}\), how many elements of (\mathcal{P}(A)) do not contain (p)?
#sets
#power-set
#counting
#excluded-element
A (8)
B (4)
C (12)
D (15)
Explanation opens after your attempt
Step 1
Concept
Excluding (p), subsets are formed only from (q,r,s), so \(2^3=8\). When one element is forbidden, take the power of the remaining elements.
Step 2
Why this answer is correct
The correct answer is A. (8). Excluding (p), subsets are formed only from (q,r,s), so \(2^3=8\). When one element is forbidden, take the power of the remaining elements.
Step 3
Exam Tip
(p) को बाहर रखकर केवल (q,r,s) से उपसमुच्चय बनेंगे, इसलिए \(2^3=8\)। किसी तत्व को हटाने पर शेष तत्वों की शक्ति लें।
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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 \(A=\{p,q,r,s,t\}\) 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 (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 \(A=\{1,2,3,4,5,6\}\), तो ऐसे उपसमुच्चयों की संख्या कितनी है जिनमें 2 नहीं हो?
If \(A=\{1,2,3,4,5,6\}\), how many subsets do not contain 2?
#subset counting
#excluded element
A 16
B 32
C 48
D 64
Explanation opens after your attempt
Step 1
Concept
After excluding 2, 5 elements remain. Their subsets are \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is B. 32. After excluding 2, 5 elements remain. Their subsets are \(2^5=32\).
Step 3
Exam Tip
2 को हटाने के बाद 5 अवयव बचते हैं। इनके सभी उपसमुच्चय \(2^5=32\) हैं।
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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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