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

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Question 501/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(\sum_{r=0}^{n}{}^{n}C_r{}^{n-r}C_m\) का सही सरल रूप कौन-सा है?

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

Explanation opens after your attempt
Correct Answer

A. \(^{n}C_m2^{n-m}\)

Explanation

Simple Explanation

पहले (m) विशेष सदस्यों को दूसरे भाग में रखिए और बाकी (n-m) सदस्यों को स्वतंत्र विकल्प दीजिए। परीक्षा में ऐसे sums में fixed marked set पहले चुनें। / First place the (m) special members in the second part and give free choices to the remaining (n-m) members. In exams choose the fixed marked set first in such sums.

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Question 502/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_3\) को किस रूप में लिखा जाएगा?

In which form is \(\sum_{r=0}^{n}{}^{n}C_r{}^{r}C_3\) written?

Explanation opens after your attempt
Correct Answer

B. \(^{n}C_3 2^{n-3}\)

Explanation

Simple Explanation

पहले (3) चिह्नित सदस्य चुनें और बाकी (n-3) सदस्य subset में आएं या न आएं। परीक्षा में अंदर वाले चयन को पहले गिनना तेज होता है। / Choose the (3) marked members first and let each of the remaining (n-3) members enter the subset or not. In exams count the inner selection first.

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Question 503/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(\sum_{r=0}^{n}r{}^{n}C_r x^r\) किस derivative identity से जुड़ा है?

Which derivative identity is connected with \(\sum_{r=0}^{n}r{}^{n}C_r x^r\)?

Explanation opens after your attempt
Correct Answer

A. (nx(1+x)^{n-1})

Explanation

Simple Explanation

((1+x)^n) को differentiate करके (x) से multiply करने पर यह sum मिलता है। परीक्षा में (r) factor दिखे तो derivative method सोचें। / Differentiate ((1+x)^n) and multiply by (x) to obtain this sum. In exams think of the derivative method when a factor (r) appears.

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Question 504/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(\sum_{r=0}^{n}r(r-1){}^{n}C_r x^r) का सही रूप कौन-सा है?

What is the correct form of (\sum_{r=0}^{n}r(r-1){}^{n}C_r x^r)?

Explanation opens after your attempt
Correct Answer

A. (n(n-1)x-2(1+x)^{n-2})

Explanation

Simple Explanation

दो बार differentiation करने पर (r(r-1)) factor आता है और \(x^2\) से power restore होती है। परीक्षा में (r(r-1)) के लिए second derivative लगाएं। / Two differentiations produce the factor (r(r-1)), and \(x^2\) restores the power. In exams use the second derivative for (r(r-1)).

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Question 505/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(\sum_{r=0}^{n}r^3{}^{n}C_r\) निकालने में कौन-सा decomposition सबसे उपयोगी है?

Which decomposition is most useful for evaluating \(\sum_{r=0}^{n}r^3{}^{n}C_r\)?

Explanation opens after your attempt
Correct Answer

A. (r-3=r(r-1)(r-2)+3r(r-1)+r)

Explanation

Simple Explanation

Powers को falling factorials में तोड़ने से standard binomial sums लगते हैं। परीक्षा में \(r^3\) को सीधे expand करने के बजाय falling form लिखें। / Breaking powers into falling factorials allows standard binomial sums. In exams write \(r^3\) in falling form instead of expanding directly.

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Question 506/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\({}^{n}C_r{}^{r}C_s={}^{n}C_s{}^{n-s}C_{r-s}\) का मुख्य कारण क्या है?

What is the main reason for \({}^{n}C_r{}^{r}C_s={}^{n}C_s{}^{n-s}C_{r-s}\)?

Explanation opens after your attempt
Correct Answer

A. पहले बड़ा समूह चुनना या पहले marked (s)-समूह चुनना एक ही कार्य हैChoosing the large group first or choosing the marked (s)-group first is the same task

Explanation

Simple Explanation

दोनों तरफ (r)-समूह के भीतर (s) विशेष सदस्यों वाला same selection गिना जाता है। परीक्षा में nested selection को order बदलकर देखें। / Both sides count the same selection with (s) special members inside an (r)-group. In exams change the order of nested selection.

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Question 507/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\({}^{n}C_a{}^{n-a}C_b{}^{n-a-b}C_c\) किस factorial form में बदलेगा?

Into which factorial form does \({}^{n}C_a{}^{n-a}C_b{}^{n-a-b}C_c\) convert?

Explanation opens after your attempt
Correct Answer

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

Explanation

Simple Explanation

Sequential selection में हर selected block का internal order हटता है। परीक्षा में कई labelled groups दिखें तो multinomial denominator बनाएं। / In sequential selection the internal order of each selected block is removed. In exams form a multinomial denominator when many labelled groups appear.

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Question 508/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि (12) distinct objects को (2,2,4,4) आकार के unlabelled groups में बांटना हो, तो extra division कौन-सी होगी?

If (12) distinct objects are divided into unlabelled groups of sizes (2,2,4,4), what is the extra division?

Explanation opens after your attempt
Correct Answer

A. \(2!\cdot2!\)

Explanation

Simple Explanation

दो size (2) groups और दो size (4) groups की अदला-बदली duplicate देती है। परीक्षा में equal-size unlabelled groups के factorials से extra divide करें। / Interchanging the two size (2) groups and the two size (4) groups gives duplicates. In exams divide extra by factorials of equal-size unlabelled groups.

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Question 509/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(15) distinct objects को (5,5,5) के unlabelled groups में बांटने का formula कौन-सा है?

What is the formula for dividing (15) distinct objects into unlabelled groups of (5,5,5)?

Explanation opens after your attempt
Correct Answer

B. (\frac{15!}{(5!)3 3!})

Explanation

Simple Explanation

तीनों groups same size और unlabelled हैं इसलिए group order (3!) भी हटता है। परीक्षा में equal unlabelled groups में extra (3!) याद रखें। / All three groups have the same size and are unlabelled, so group order (3!) is also removed. In exams remember the extra (3!) for equal unlabelled groups.

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Question 510/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

कौन-सा formula (n) distinct objects को (r) labelled boxes में exactly (k) empty boxes के साथ distribute करता है?

Which formula distributes (n) distinct objects into (r) labelled boxes with exactly (k) empty boxes?

Explanation opens after your attempt
Correct Answer

A. (^{r}C_k\sum_{i=0}^{r-k}(-1)^i{}^{r-k}C_i(r-k-i)^n)

Explanation

Simple Explanation

पहले empty boxes चुनें, फिर बाकी boxes में onto distribution करें। परीक्षा में exactly empty boxes में selection plus inclusion-exclusion लगाएं। / First choose the empty boxes, then distribute onto the remaining boxes. In exams use selection plus inclusion-exclusion for exactly empty boxes.

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