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Mathematics Derivations of formulas and their connections MCQ Questions for Class 11 General

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Question 401/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}P_r\) को \(^{n}P_{r-1}\) से निकाला जाए, तो कौन-सा गुणक जुड़ता है?

If \(^{n}P_r\) is obtained from \(^{n}P_{r-1}\), which multiplier is added?

Explanation opens after your attempt
Correct Answer

B. (n-r+1)

Explanation

Simple Explanation

एक और स्थान भरने पर उपलब्ध विकल्प (n-r+1) होते हैं। परीक्षा में क्रमचय के अगले पद में नया अंतिम factor देखें। / Filling one more position gives (n-r+1) choices. In exams identify the new last factor in the next permutation term.

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Question 402/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

संबंध \(^{n}C_r=\frac{n}{r}{}^{n-1}C_{r-1}\) किस विचार से समझा जाता है?

Which idea explains the relation \(^{n}C_r=\frac{n}{r}{}^{n-1}C_{r-1}\)?

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Correct Answer

A. एक चुने हुए सदस्य को चिह्नित करके गिननाCounting by marking one selected member

Explanation

Simple Explanation

पहले चिह्नित सदस्य चुनने पर (n) विकल्प आते हैं और हर समूह (r) बार गिना जाता है। परीक्षा में (r) वाला denominator overcount समझें। / Choosing the marked member first gives (n) choices and each group is counted (r) times. In exams treat the denominator (r) as overcount.

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Question 403/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}C_r\) को \(^{n-1}C_{r-1}\) से तुलना करें, तो सही अनुपात कौन-सा है?

If \(^{n}C_r\) is compared with \(^{n-1}C_{r-1}\), which ratio is correct?

Explanation opens after your attempt
Correct Answer

C. \(\frac{{}^{n}C_r}{{}^{n-1}C_{r-1}}=\frac{n}{r}\)

Explanation

Simple Explanation

Factorial रूप cancel करने पर \(\frac{n}{r}\) मिलता है। परीक्षा में adjacent upper और lower indices में ratio method उपयोग करें। / Canceling factorial forms gives \(\frac{n}{r}\). In exams use the ratio method for adjacent upper and lower indices.

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Question 404/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}C_r+{}^{n}C_{r-1}\) को Pascal identity से किस रूप में लिखा जाएगा?

Using Pascal identity, \(^{n}C_r+{}^{n}C_{r-1}\) is written in which form?

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Correct Answer

A. \(^{n+1}C_r\)

Explanation

Simple Explanation

एक ही पंक्ति के adjacent terms अगली पंक्ति का बीच वाला term देते हैं। परीक्षा में (r) और (r-1) साथ दिखें तो Pascal identity लगाएं। / Adjacent terms of the same row give the middle term in the next row. In exams apply Pascal identity when (r) and (r-1) appear together.

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Question 405/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{15}C_{r}=^{15}C_{r+3}\) और indices अलग हैं, तो (r) का मान क्या होगा?

If \(^{15}C_{r}=^{15}C_{r+3}\) and the indices are different, what is the value of (r)?

Explanation opens after your attempt
Correct Answer

B. (6)

Explanation

Simple Explanation

Unequal equal-combination indices complementary होते हैं, इसलिए (r+r+3=15)। परीक्षा में lower indices का sum upper index के बराबर रखें। / Unequal equal-combination indices are complementary, so (r+r+3=15). In exams set the sum of lower indices equal to the upper index.

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Question 406/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{18}C_{2r}=^{18}C_{r+3}\) और \(2r\neq r+3\) हो, तो (r) क्या होगा?

If \(^{18}C_{2r}=^{18}C_{r+3}\) and \(2r\neq r+3\), what is (r)?

Explanation opens after your attempt
Correct Answer

C. (5)

Explanation

Simple Explanation

Complementary condition से (2r+r+3=18), इसलिए (r=5)। परीक्षा में equality के दो cases सोचें: same index या complementary index। / The complementary condition gives (2r+r+3=18), so (r=5). In exams consider two cases for equality: same index or complementary index.

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Question 407/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}C_r=^{n}C_s\) में यदि \(r\neq s\), तो कौन-सी शर्त सही होती है?

In \(^{n}C_r=^{n}C_s\), if \(r\neq s\), which condition is correct?

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Correct Answer

A. (r+s=n)

Explanation

Simple Explanation

समान मान वाले अलग indices complementary होते हैं। परीक्षा में ऐसे सवालों में indices जोड़कर (n) से मिलाएं। / Different indices with equal values are complementary. In exams add the indices and match them with (n).

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Question 408/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}P_r=^{n}C_r\cdot r!\) में (r!) किस चीज़ को दर्शाता है?

In \(^{n}P_r=^{n}C_r\cdot r!\), what does (r!) represent?

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Correct Answer

A. चुनी गई (r) वस्तुओं के सभी क्रमAll orders of the selected (r) objects

Explanation

Simple Explanation

Combination केवल समूह देता है और (r!) उस समूह के arrangements जोड़ता है। परीक्षा में permutation को selection plus arrangement समझें। / Combination gives only the group and (r!) adds arrangements of that group. In exams treat permutation as selection plus arrangement.

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Question 409/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}C_r=\frac{^{n}P_r}{r!}\) का मुख्य कारण क्या है?

What is the main reason for \(^{n}C_r=\frac{^{n}P_r}{r!}\)?

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Correct Answer

A. Permutation में order counted होता है और combination में नहींOrder is counted in permutation but not in combination

Explanation

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Ordered count से unordered count पाने के लिए (r!) orders हटाए जाते हैं। परीक्षा में order ignored हो तो divide करें। / To get unordered count from ordered count, (r!) orders are removed. In exams divide when order is ignored.

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Question 410/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(n) अलग वस्तुओं में से कम से कम (3) वस्तुएं चुनने की संख्या कौन-सी है?

What is the number of ways to select at least (3) objects from (n) distinct objects?

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Correct Answer

A. \(2^n-1-n-{}^{n}C_2\)

Explanation

Simple Explanation

सभी subsets में से (0), (1), और (2) selections घटाते हैं। परीक्षा में at least को total minus small unwanted cases से करें। / Subtract selections of (0), (1), and (2) from all subsets. In exams handle at least by total minus small unwanted cases.

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