(9) पुस्तकों में से (3) पुस्तकों का सेट कितने तरीकों से बनाया जा सकता है?
In how many ways can a set of (3) books be made from (9) books?
Explanation opens after your attempt
Step 1
Concept
Order is not important while making a set. Hence \(\binom{9}{3}=84\) is correct.
Step 2
Why this answer is correct
The correct answer is C. (84). Order is not important while making a set. Hence \(\binom{9}{3}=84\) is correct.
Step 3
Exam Tip
सेट बनाने में क्रम का महत्व नहीं होता। इसलिए \(\binom{9}{3}=84\) सही है।
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(8) विद्यार्थियों में से (6) विद्यार्थियों की टीम कितने तरीकों से चुनी जा सकती है?
In how many ways can a team of (6) students be selected from (8) students?
Explanation opens after your attempt
Step 1
Concept
\(\binom{8}{6}=\binom{8}{2}=28\). When many are selected, counting those not selected is easier.
Step 2
Why this answer is correct
The correct answer is B. (28). \(\binom{8}{6}=\binom{8}{2}=28\). When many are selected, counting those not selected is easier.
Step 3
Exam Tip
\(\binom{8}{6}=\binom{8}{2}=28\) होता है। अधिक चुनना हो तो न चुने गए लोगों को गिनना आसान होता है।
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(15) विद्यार्थियों में से (2) विद्यार्थियों की जोड़ी चुनने के कितने तरीके हैं?
How many ways are there to select a pair of (2) students from (15) students?
Explanation opens after your attempt
Step 1
Concept
The number of ways is \(\binom{15}{2}=105\). Order is not counted in a pair.
Step 2
Why this answer is correct
The correct answer is A. (105). The number of ways is \(\binom{15}{2}=105\). Order is not counted in a pair.
Step 3
Exam Tip
जोड़ी चुनने के तरीके \(\binom{15}{2}=105\) हैं। जोड़ी में क्रम नहीं गिना जाता।
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(7) रंगीन पेंसिलों में से कम से कम (1) पेंसिल चुनने के कितने तरीके हैं?
In how many ways can at least (1) pencil be selected from (7) colored pencils?
Explanation opens after your attempt
Step 1
Concept
Total subsets are \(2^7\), and the empty selection is removed. Hence the ways are \(2^7-1=127\).
Step 2
Why this answer is correct
The correct answer is C. (127). Total subsets are \(2^7\), and the empty selection is removed. Hence the ways are \(2^7-1=127\).
Step 3
Exam Tip
कुल उपसमुच्चय \(2^7\) हैं और खाली चयन हटेगा। इसलिए \(2^7-1=127\) तरीके हैं।
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(12) खिलौनों में से (4) खिलौने चुनने के कितने तरीके हैं?
How many ways are there to choose (4) toys from (12) toys?
Explanation opens after your attempt
Step 1
Concept
This is \(\binom{12}{4}=495\). For larger values, simplify the formula step by step.
Step 2
Why this answer is correct
The correct answer is C. (495). This is \(\binom{12}{4}=495\). For larger values, simplify the formula step by step.
Step 3
Exam Tip
यह \(\binom{12}{4}=495\) है। बड़े मानों में सूत्र को क्रम से सरल करें।
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(5) लड़कों और (5) लड़कियों में से (1) लड़का और (1) लड़की चुनने के कितने तरीके हैं?
From (5) boys and (5) girls, in how many ways can (1) boy and (1) girl be selected?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{5}{1}\binom{5}{1}=25\). The two selections are independent, so multiply.
Step 2
Why this answer is correct
The correct answer is C. (25). The ways are \(\binom{5}{1}\binom{5}{1}=25\). The two selections are independent, so multiply.
Step 3
Exam Tip
तरीके \(\binom{5}{1}\binom{5}{1}=25\) हैं। दोनों चयन स्वतंत्र हैं इसलिए गुणा करें।
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(8) अक्षरों में से (3) अक्षरों का चयन कितने तरीकों से हो सकता है यदि क्रम महत्वपूर्ण नहीं है?
In how many ways can (3) letters be selected from (8) letters if order is not important?
Explanation opens after your attempt
Step 1
Concept
Order is not important, so \(\binom{8}{3}=56\). Use combinations in selection questions.
Step 2
Why this answer is correct
The correct answer is B. (56). Order is not important, so \(\binom{8}{3}=56\). Use combinations in selection questions.
Step 3
Exam Tip
क्रम महत्वपूर्ण नहीं है इसलिए \(\binom{8}{3}=56\) है। चयन वाले प्रश्नों में संयोजन लगाएं।
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(10) बिंदुओं से कितने रेखाखंड बनाए जा सकते हैं?
How many line segments can be formed from (10) points?
Explanation opens after your attempt
Step 1
Concept
A line segment needs (2) points. Therefore \(\binom{10}{2}=45\).
Step 2
Why this answer is correct
The correct answer is B. (45). A line segment needs (2) points. Therefore \(\binom{10}{2}=45\).
Step 3
Exam Tip
एक रेखाखंड के लिए (2) बिंदु चाहिए। इसलिए \(\binom{10}{2}=45\) है।
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(6) अंकों में से (2) अंक चुनने के कितने तरीके हैं यदि क्रम नहीं देखना है?
How many ways are there to choose (2) digits from (6) digits if order is not considered?
Explanation opens after your attempt
Step 1
Concept
Order is not considered, so \(\binom{6}{2}=15\). This is the main difference between combination and permutation.
Step 2
Why this answer is correct
The correct answer is B. (15). Order is not considered, so \(\binom{6}{2}=15\). This is the main difference between combination and permutation.
Step 3
Exam Tip
क्रम नहीं देखना है इसलिए \(\binom{6}{2}=15\) है। यही संयोजन और क्रमचय का मुख्य अंतर है।
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(8) अलग-अलग चित्रों में से (3) चित्र प्रदर्शनी के लिए चुनने के कितने तरीके हैं?
How many ways are there to choose (3) pictures from (8) different pictures for an exhibition?
Explanation opens after your attempt
Step 1
Concept
The selection is \(\binom{8}{3}=56\). Choosing for an exhibition is a combination.
Step 2
Why this answer is correct
The correct answer is B. (56). The selection is \(\binom{8}{3}=56\). Choosing for an exhibition is a combination.
Step 3
Exam Tip
चयन \(\binom{8}{3}=56\) होगा। प्रदर्शनी के लिए चुनना संयोजन है।
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(9) गीतों में से (4) गीत चुनने के कितने तरीके हैं?
How many ways are there to choose (4) songs from (9) songs?
Explanation opens after your attempt
Step 1
Concept
Choosing songs is not arrangement, so \(\binom{9}{4}=126\). If order were needed, it would be a different question.
Step 2
Why this answer is correct
The correct answer is C. (126). Choosing songs is not arrangement, so \(\binom{9}{4}=126\). If order were needed, it would be a different question.
Step 3
Exam Tip
गीत चुनना व्यवस्था नहीं है इसलिए \(\binom{9}{4}=126\) है। क्रम चाहिए होता तो अलग प्रश्न होता।
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(7) अध्यापकों में से (2) अध्यापकों की निगरानी टीम कितने तरीकों से चुनी जा सकती है?
In how many ways can an invigilation team of (2) teachers be selected from (7) teachers?
Explanation opens after your attempt
Step 1
Concept
For the team, there are \(\binom{7}{2}=21\) selections. A team has no order.
Step 2
Why this answer is correct
The correct answer is B. (21). For the team, there are \(\binom{7}{2}=21\) selections. A team has no order.
Step 3
Exam Tip
टीम के लिए \(\binom{7}{2}=21\) चयन होंगे। टीम में क्रम नहीं होता।
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(5) व्यंजनों में से (2) व्यंजन चुनकर थाली बनानी है। कितनी थालियां बन सकती हैं?
A plate is to be made by choosing (2) dishes from (5) dishes. How many plates can be made?
Explanation opens after your attempt
Step 1
Concept
The ways to choose two dishes are \(\binom{5}{2}=10\). The order on the plate is not important.
Step 2
Why this answer is correct
The correct answer is A. (10). The ways to choose two dishes are \(\binom{5}{2}=10\). The order on the plate is not important.
Step 3
Exam Tip
दो व्यंजन चुनने के तरीके \(\binom{5}{2}=10\) हैं। थाली में क्रम का महत्व नहीं है।
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(10) अलग-अलग प्रश्नों में से (3) प्रश्नों का सेट कितने तरीकों से बनाया जा सकता है?
In how many ways can a set of (3) questions be made from (10) different questions?
Explanation opens after your attempt
Step 1
Concept
Making a set of questions is selection. Hence \(\binom{10}{3}=120\) is correct.
Step 2
Why this answer is correct
The correct answer is C. (120). Making a set of questions is selection. Hence \(\binom{10}{3}=120\) is correct.
Step 3
Exam Tip
प्रश्नों का सेट बनाना चयन है। इसलिए \(\binom{10}{3}=120\) सही है।
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(6) विद्यार्थियों में से (5) विद्यार्थियों को प्रतियोगिता के लिए चुनने के कितने तरीके हैं?
How many ways are there to select (5) students from (6) students for a competition?
Explanation opens after your attempt
Step 1
Concept
\(\binom{6}{5}=\binom{6}{1}=6\). Sometimes counting the person not selected is easier.
Step 2
Why this answer is correct
The correct answer is B. (6). \(\binom{6}{5}=\binom{6}{1}=6\). Sometimes counting the person not selected is easier.
Step 3
Exam Tip
\(\binom{6}{5}=\binom{6}{1}=6\) है। कभी-कभी न चुने गए व्यक्ति को गिनना आसान होता है।
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(8) छात्रों में से (2) छात्रों को पुस्तकालय कार्य के लिए चुनना है। कितने चयन संभव हैं?
(2) students are to be selected from (8) students for library work. How many selections are possible?
Explanation opens after your attempt
Step 1
Concept
This is \(\binom{8}{2}=28\). If the work is same, order is not counted in selection.
Step 2
Why this answer is correct
The correct answer is B. (28). This is \(\binom{8}{2}=28\). If the work is same, order is not counted in selection.
Step 3
Exam Tip
यह \(\binom{8}{2}=28\) है। कार्य समान हो तो चयन में क्रम नहीं गिना जाता।
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(4) सफेद और (5) काली गेंदों में से (2) काली गेंदें कितने तरीकों से चुनी जा सकती हैं?
From (4) white and (5) black balls, in how many ways can (2) black balls be selected?
Explanation opens after your attempt
Step 1
Concept
Only black balls are selected, so \(\binom{5}{2}=10\). The remaining information is only context.
Step 2
Why this answer is correct
The correct answer is A. (10). Only black balls are selected, so \(\binom{5}{2}=10\). The remaining information is only context.
Step 3
Exam Tip
केवल काली गेंदों में से चयन है इसलिए \(\binom{5}{2}=10\) होगा। बाकी सूचना केवल संदर्भ है।
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(5) पेन और (6) पेंसिलों में से (2) पेन और (1) पेंसिल कितने तरीकों से चुनी जा सकती हैं?
From (5) pens and (6) pencils, in how many ways can (2) pens and (1) pencil be selected?
Explanation opens after your attempt
Step 1
Concept
The ways are \(\binom{5}{2}\binom{6}{1}=10\cdot6=60\). Select from each group separately.
Step 2
Why this answer is correct
The correct answer is C. (60). The ways are \(\binom{5}{2}\binom{6}{1}=10\cdot6=60\). Select from each group separately.
Step 3
Exam Tip
तरीके \(\binom{5}{2}\binom{6}{1}=10\cdot6=60\) हैं। प्रत्येक समूह से चयन अलग-अलग करें।
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(3) गणित और (4) विज्ञान पुस्तकों में से (1) गणित और (1) विज्ञान पुस्तक कितने तरीकों से चुनी जा सकती है?
From (3) mathematics books and (4) science books, in how many ways can (1) mathematics book and (1) science book be chosen?
Explanation opens after your attempt
Step 1
Concept
The selection is \(\binom{3}{1}\binom{4}{1}=12\). Choices from different types are multiplied.
Step 2
Why this answer is correct
The correct answer is B. (12). The selection is \(\binom{3}{1}\binom{4}{1}=12\). Choices from different types are multiplied.
Step 3
Exam Tip
चयन \(\binom{3}{1}\binom{4}{1}=12\) है। अलग प्रकारों में चयन की संख्याएं गुणा होती हैं।
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\(\binom{5}{2}+\binom{5}{3}\) का मान क्या है?
What is the value of \(\binom{5}{2}+\binom{5}{3}\)?
Explanation opens after your attempt
Step 1
Concept
\(\binom{5}{2}=10\) and \(\binom{5}{3}=10\). Therefore the total is (20).
Step 2
Why this answer is correct
The correct answer is C. (20). \(\binom{5}{2}=10\) and \(\binom{5}{3}=10\). Therefore the total is (20).
Step 3
Exam Tip
\(\binom{5}{2}=10\) और \(\binom{5}{3}=10\) हैं। इसलिए कुल (20) है।
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\(\binom{6}{2}\) और \(\binom{6}{4}\) में क्या संबंध है?
What is the relation between \(\binom{6}{2}\) and \(\binom{6}{4}\)?
Explanation opens after your attempt
Correct Answer
A. दोनों बराबर हैं/They are equal
Step 1
Concept
Because \(\binom{n}{r}=\binom{n}{n-r}\). Here (r=2) and (n-r=4).
Step 2
Why this answer is correct
The correct answer is A. दोनों बराबर हैं / They are equal. Because \(\binom{n}{r}=\binom{n}{n-r}\). Here (r=2) and (n-r=4).
Step 3
Exam Tip
क्योंकि \(\binom{n}{r}=\binom{n}{n-r}\) होता है। यहां (r=2) और (n-r=4) हैं।
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यदि \(\binom{n}{0}=1\) है तो यह किस (n) के लिए सत्य है?
If \(\binom{n}{0}=1\), for which (n) is it true?
Explanation opens after your attempt
Correct Answer
C. हर \(n\geq0\)/Every \(n\geq0\)
Step 1
Concept
There is one way to choose zero objects for every \(n\geq0\). This is a basic rule.
Step 2
Why this answer is correct
The correct answer is C. हर \(n\geq0\) / Every \(n\geq0\). There is one way to choose zero objects for every \(n\geq0\). This is a basic rule.
Step 3
Exam Tip
शून्य वस्तु चुनने का एक तरीका हर \(n\geq0\) के लिए होता है। यह मूल नियम है।
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यदि \(\binom{n}{1}=14\) है तो (n) का मान क्या है?
If \(\binom{n}{1}=14\), what is the value of (n)?
Explanation opens after your attempt
Step 1
Concept
\(\binom{n}{1}=n\). Therefore (n=14).
Step 2
Why this answer is correct
The correct answer is C. (14). \(\binom{n}{1}=n\). Therefore (n=14).
Step 3
Exam Tip
\(\binom{n}{1}=n\) होता है। इसलिए (n=14) मिलेगा।
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\(\binom{11}{1}+\binom{11}{0}\) का मान क्या है?
What is the value of \(\binom{11}{1}+\binom{11}{0}\)?
Explanation opens after your attempt
Step 1
Concept
The values are \(\binom{11}{1}=11\) and \(\binom{11}{0}=1\). The total is (12).
Step 2
Why this answer is correct
The correct answer is C. (12). The values are \(\binom{11}{1}=11\) and \(\binom{11}{0}=1\). The total is (12).
Step 3
Exam Tip
मान \(\binom{11}{1}=11\) और \(\binom{11}{0}=1\) हैं। कुल (12) होगा।
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\(\binom{9}{2}\) का मान ज्ञात कीजिए।
Find the value of \(\binom{9}{2}\).
Explanation opens after your attempt
Step 1
Concept
\(\binom{9}{2}=\frac{9\cdot8}{2}=36\). Do not forget to divide by (2!) when choosing two objects.
Step 2
Why this answer is correct
The correct answer is B. (36). \(\binom{9}{2}=\frac{9\cdot8}{2}=36\). Do not forget to divide by (2!) when choosing two objects.
Step 3
Exam Tip
\(\binom{9}{2}=\frac{9\cdot8}{2}=36\) है। दो वस्तुओं के चयन में (2!) से भाग देना न भूलें।
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\(\binom{12}{1}\) का मान क्या है?
What is the value of \(\binom{12}{1}\)?
Explanation opens after your attempt
Step 1
Concept
\(\binom{n}{1}=n\). Hence \(\binom{12}{1}=12\) is correct.
Step 2
Why this answer is correct
The correct answer is C. (12). \(\binom{n}{1}=n\). Hence \(\binom{12}{1}=12\) is correct.
Step 3
Exam Tip
\(\binom{n}{1}=n\) होता है। इसलिए \(\binom{12}{1}=12\) सही है।
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\(\binom{7}{7}\) का मान क्या है?
What is the value of \(\binom{7}{7}\)?
Explanation opens after your attempt
Step 1
Concept
There is only one way to choose all (7) objects. Therefore \(\binom{7}{7}=1\).
Step 2
Why this answer is correct
The correct answer is B. (1). There is only one way to choose all (7) objects. Therefore \(\binom{7}{7}=1\).
Step 3
Exam Tip
सभी (7) वस्तुओं को चुनने का केवल एक तरीका है। अतः \(\binom{7}{7}=1\) है।
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\(\binom{9}{0}\) का मान क्या होगा?
What will be the value of \(\binom{9}{0}\)?
Explanation opens after your attempt
Step 1
Concept
For any (n), \(\binom{n}{0}=1\). This is a basic identity of combinations.
Step 2
Why this answer is correct
The correct answer is B. (1). For any (n), \(\binom{n}{0}=1\). This is a basic identity of combinations.
Step 3
Exam Tip
किसी भी (n) के लिए \(\binom{n}{0}=1\) होता है। यह संयोजन की मूल पहचान है।
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\(\binom{10}{3}\) का मान निकालिए।
Find the value of \(\binom{10}{3}\).
Explanation opens after your attempt
Step 1
Concept
\(\binom{10}{3}=\frac{10\cdot9\cdot8}{3\cdot2\cdot1}=120\). For small (r), multiply directly.
Step 2
Why this answer is correct
The correct answer is C. (120). \(\binom{10}{3}=\frac{10\cdot9\cdot8}{3\cdot2\cdot1}=120\). For small (r), multiply directly.
Step 3
Exam Tip
\(\binom{10}{3}=\frac{10\cdot9\cdot8}{3\cdot2\cdot1}=120\) है। छोटे (r) के लिए सीधे गुणा करें।
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\(\binom{8}{2}\) का मान क्या है?
What is the value of \(\binom{8}{2}\)?
Explanation opens after your attempt
Step 1
Concept
\(\binom{8}{2}=\frac{8\cdot7}{2}=28\). The formula for choosing two objects is quick.
Step 2
Why this answer is correct
The correct answer is C. (28). \(\binom{8}{2}=\frac{8\cdot7}{2}=28\). The formula for choosing two objects is quick.
Step 3
Exam Tip
\(\binom{8}{2}=\frac{8\cdot7}{2}=28\) है। दो वस्तुओं के चयन का सूत्र तेज होता है।
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