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

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Question 441/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(7) symbols से repetition allowed (5)-character passwords बनते हैं। Count \(7^5\) क्यों है?

(5)-character passwords are formed from (7) symbols with repetition allowed. Why is the count \(7^5\)?

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Correct Answer

A. हर स्थान पर (7) independent choices हैंEach position has (7) independent choices

Explanation

Simple Explanation

Repetition allowed होने पर selected symbol फिर उपलब्ध रहता है। परीक्षा में independent positions में power rule लगाएं। / When repetition is allowed, a selected symbol remains available again. In exams use the power rule for independent positions.

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Question 442/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(7) symbols से repetition बिना (5)-character passwords बनते हैं। Count कौन-सी है?

(5)-character passwords are formed from (7) symbols without repetition. Which count is correct?

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Correct Answer

C. \(^{7}P_5\)

Explanation

Simple Explanation

Password में order important है और repetition नहीं है। परीक्षा में without repetition ordered slots में permutation लगाएं। / Order matters in passwords and repetition is not allowed. In exams use permutation for ordered slots without repetition.

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Question 443/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

((a+b)^n) में \(a^{n-r}b^r\) का coefficient \(^{n}C_r\) क्यों होता है?

Why is the coefficient of \(a^{n-r}b^r\) in ((a+b)^n) equal to \(^{n}C_r\)?

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Correct Answer

A. क्योंकि (r) brackets से (b) चुनते हैंBecause (b) is chosen from (r) brackets

Explanation

Simple Explanation

\(b^r\) बनाने के लिए (n) brackets में से (r) brackets चुनते हैं। परीक्षा में binomial coefficient को bracket selection से जोड़ें। / To form \(b^r\), choose (r) brackets from (n) brackets. In exams connect binomial coefficients with bracket selection.

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Question 444/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

((1+x)^{10}) में \(x^4\) का coefficient कौन-सा है?

What is the coefficient of \(x^4\) in ((1+x)^{10})?

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Correct Answer

B. \(^{10}C_4\)

Explanation

Simple Explanation

(10) brackets में से (4) brackets से (x) चुनना होता है। परीक्षा में coefficient के लिए order नहीं गिनें। / Choose (x) from (4) of the (10) brackets. In exams do not count order for coefficients.

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Question 445/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}{}^{n}C_r=2^n\) को subsets से कैसे समझते हैं?

How is \(\sum_{r=0}^{n}{}^{n}C_r=2^n\) understood through subsets?

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Correct Answer

A. सभी possible subset sizes को जोड़नाAdding all possible subset sizes

Explanation

Simple Explanation

Left side हर size के subsets को जोड़ता है और right side प्रत्येक object के choose or not choose choices देता है। परीक्षा में subset identity को दो तरीकों से गिनें। / The left side adds subsets of every size and the right side gives choose-or-not choices for each object. In exams count subset identities in two ways.

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Question 446/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(\sum_{r=0}^{n}(-1)^r{}^{n}C_r=0) के लिए कौन-सा substitution उपयोग होता है?

Which substitution is used for (\sum_{r=0}^{n}(-1)^r{}^{n}C_r=0)?

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Correct Answer

B. ((1+x)^n) में (x=-1)(x=-1) in ((1+x)^n)

Explanation

Simple Explanation

(x=-1) रखने पर ((1-1)^n=0) मिलता है। परीक्षा में alternating sum में (x=-1) याद रखें। / Putting (x=-1) gives ((1-1)^n=0). In exams remember (x=-1) for alternating sums.

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Question 447/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}r{}^{n}C_r=n2^{n-1}\) किस double counting से आता है?

Which double counting gives \(\sum_{r=0}^{n}r{}^{n}C_r=n2^{n-1}\)?

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Correct Answer

A. एक selected सदस्य को mark करनाMarking one selected member

Explanation

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पहले subset चुनकर उसमें एक member mark करें या पहले marked member चुनकर बाकी freely चुनें। परीक्षा में (r) factor को marked choice समझें। / Choose a subset and mark one member, or first choose the marked member and freely choose the rest. In exams treat the factor (r) as a marked choice.

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Question 448/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_2\) का simplified form क्या है?

What is the simplified form of \(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_2\)?

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Correct Answer

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

Explanation

Simple Explanation

पहले marked pair चुनें, फिर बाकी (n-2) objects freely choose करें। परीक्षा में nested combination sums में marked objects पहले चुनें। / First choose the marked pair, then freely choose from the remaining (n-2) objects. In exams choose marked objects first in nested combination sums.

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Question 449/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\({}^{m+n}C_r=\sum_{k=0}^{r}{}^{m}C_k{}^{n}C_{r-k}\) किस identity का रूप है?

The identity \({}^{m+n}C_r=\sum_{k=0}^{r}{}^{m}C_k{}^{n}C_{r-k}\) is a form of which identity?

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Correct Answer

B. Vandermonde identity

Explanation

Simple Explanation

दो groups से कुल (r) objects चुनने में first group से (k) objects लिए जाते हैं। परीक्षा में two-group selection में Vandermonde पहचानें। / When choosing (r) objects from two groups, (k) objects are taken from the first group. In exams identify Vandermonde in two-group selection.

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Question 450/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\({}^{n}C_a{}^{n-a}C_b\) को किस factorial form से जोड़ा जा सकता है?

With which factorial form can \({}^{n}C_a{}^{n-a}C_b\) be connected?

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Correct Answer

A. (\frac{n!}{a!b!(n-a-b)!})

Explanation

Simple Explanation

पहले (a) objects और फिर बचे हुए से (b) objects चुनना labelled grouping जैसा है। परीक्षा में sequential selections को factorial division में बदलें। / Choosing (a) objects first and then (b) from the remaining objects is like labelled grouping. In exams convert sequential selections into factorial division.

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