यदि \(A=\{1,2,3,4,5,6\}\), \(B=\{1,2,3,4,5,6\}\) और \(R=\{(a,b):a+b=7\}\) है, तो (R) के कितने उपसमुच्चय कम से कम एक युग्म रखते हैं जिसका पहला अवयव (1) है?
There are (6) pairs in (R), and only ((1,6)) has first component (1). It must be included, so the other (5) pairs are free, giving \(2^5=32\).
Step 2
Why this answer is correct
The correct answer is B. (32). There are (6) pairs in (R), and only ((1,6)) has first component (1). It must be included, so the other (5) pairs are free, giving \(2^5=32\).
Step 3
Exam Tip
(R) में (6) युग्म हैं और पहला अवयव (1) वाला केवल ((1,6)) है। उसे रखना होगा, इसलिए बाकी (5) युग्म स्वतंत्र हैं और संख्या \(2^5=32\) है।
\(|A\times B|=10\), so choosing exactly (5) pairs gives \(\binom{10}{5}=252\). Use combinations for exact-size subsets.
Step 2
Why this answer is correct
The correct answer is C. (252). \(|A\times B|=10\), so choosing exactly (5) pairs gives \(\binom{10}{5}=252\). Use combinations for exact-size subsets.
Step 3
Exam Tip
\(|A\times B|=10\), इसलिए ठीक (5) युग्म चुनने के तरीके \(\binom{10}{5}=252\) हैं। ठीक संख्या के लिए संयोजन लगाएँ।
A subset containing ({1,2}) has (3) and (4) as optional elements, so \(2^2=4\). In exams, treat the required subset as fixed.
Step 2
Why this answer is correct
The correct answer is B. (4). A subset containing ({1,2}) has (3) and (4) as optional elements, so \(2^2=4\). In exams, treat the required subset as fixed.
Step 3
Exam Tip
जिस subset में ({1,2}) शामिल हो, उसमें (3) और (4) वैकल्पिक हैं, इसलिए \(2^2=4\)। परीक्षा में required subset को fixed block मानें।
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।
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।
(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।
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।
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।
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।
A four-element subset of (A) must take all four elements.
Step 2
Why this answer is correct
So there is only one such subset, (A) itself.
Step 3
Exam Tip
A set itself is also counted as its subset. चरण 1: चार सदस्य वाला उपसमुच्चय मूल समुच्चय के सभी चार सदस्य लेगा। चरण 2: इसलिए ऐसा केवल एक उपसमुच्चय है, यानी (A) स्वयं। चरण 3: सभी सदस्य लेने पर समुच्चय खुद भी अपना उपसमुच्चय होता है।
