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

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Derivations of formulas and their connections Practice Questions

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Question 561/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}r^2{}^{n}C_r\) का simplified result क्या है?

What is the simplified result of \(\sum_{r=0}^{n}r^2{}^{n}C_r\)?

Explanation opens after your attempt
Correct Answer

C. (n(n+1)2^{n-2})

Explanation

Simple Explanation

(r-2=r(r-1)+r) लगाने पर दोनों standard sums जुड़ते हैं। परीक्षा में final form को (n(n+1)2^{n-2}) तक simplify करें। / Using (r-2=r(r-1)+r) adds two standard sums. In exams simplify the final form to (n(n+1)2^{n-2}).

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Question 562/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(\sum_{r=0}^{n}r^3{}^{n}C_r\) के लिए कौन-सा final form सही है?

Which final form is correct for \(\sum_{r=0}^{n}r^3{}^{n}C_r\)?

Explanation opens after your attempt
Correct Answer

A. (n-2(n+3)2^{n-3})

Explanation

Simple Explanation

(r-3=r(r-1)(r-2)+3r(r-1)+r) से यह form मिलता है। परीक्षा में cubic sums में falling factorial decomposition लगाएं। / Using (r-3=r(r-1)(r-2)+3r(r-1)+r) gives this form. In exams use falling factorial decomposition for cubic sums.

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Question 563/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि (n) distinct objects को (a,a,b) आकार के unlabelled groups में बांटा जाए और (2a+b=n), तो extra division किससे होगा?

If (n) distinct objects are divided into unlabelled groups of sizes (a,a,b) and (2a+b=n), what is the extra division?

Explanation opens after your attempt
Correct Answer

C. (2!)

Explanation

Simple Explanation

दो groups का size (a) समान है इसलिए उनकी अदला-बदली duplicate देती है। परीक्षा में equal-size unlabelled groups पर extra factorial divide करें। / Two groups have the same size (a), so interchanging them gives duplicates. In exams divide extra by factorial for equal-size unlabelled groups.

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Question 564/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(14) distinct objects को (3,3,4,4) के unlabelled groups में बांटने का denominator कौन-सा होगा?

What is the denominator for dividing (14) distinct objects into unlabelled groups of sizes (3,3,4,4)?

Explanation opens after your attempt
Correct Answer

A. (3!3!4!4!2!2!)

Explanation

Simple Explanation

Internal orders और equal-size group swaps दोनों हटते हैं। परीक्षा में denominator में group sizes और equal-group factorials दोनों लिखें। / Both internal orders and equal-size group swaps are removed. In exams write both group sizes and equal-group factorials in the denominator.

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Question 565/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(n) distinct objects को (r) labelled boxes में exactly (r-1) non-empty boxes में भेजने की count कौन-सी है?

What is the count for sending (n) distinct objects into (r) labelled boxes with exactly (r-1) non-empty boxes?

Explanation opens after your attempt
Correct Answer

A. (^{r}C_1\left((r-1)^n-{}^{r-1}C_1(r-2)^n+\cdots\right))

Explanation

Simple Explanation

पहले empty box चुनें और बाकी (r-1) boxes में onto distribution करें। परीक्षा में exactly non-empty boxes में choose empty plus onto count करें। / Choose the empty box first and then distribute onto the remaining (r-1) boxes. In exams use choose empty plus onto count for exactly non-empty boxes.

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Question 566/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(n) distinct objects को (4) labelled boxes में exactly (2) boxes empty रखकर distribute करने की count क्या है?

What is the count for distributing (n) distinct objects into (4) labelled boxes with exactly (2) boxes empty?

Explanation opens after your attempt
Correct Answer

A. (^{4}C_2\(2^n-2\))

Explanation

Simple Explanation

पहले empty boxes चुनें फिर बाकी दो boxes दोनों non-empty होने चाहिए। परीक्षा में exactly empty में remaining boxes पर onto लगाएं। / Choose the empty boxes first and then the remaining two boxes must both be non-empty. In exams apply onto to the remaining boxes in exactly-empty cases.

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Question 567/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

(n) distinct objects को (3) labelled boxes में सभी boxes non-empty रखने की count क्या है?

What is the count for placing (n) distinct objects into (3) labelled boxes with all boxes non-empty?

Explanation opens after your attempt
Correct Answer

A. \(3^n-3\cdot2^n+3\)

Explanation

Simple Explanation

Inclusion-exclusion से empty box cases हटते हैं। परीक्षा में non-empty labelled boxes को onto functions की तरह गिनें। / Empty-box cases are removed by inclusion-exclusion. In exams count non-empty labelled boxes like onto functions.

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Question 568/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(x_1+x_2+x_3+x_4=22\) में \(x_1\geq1\), \(x_2\geq2\), \(x_3\geq3\), \(x_4\geq4\) हो, तो count क्या है?

In \(x_1+x_2+x_3+x_4=22\), if \(x_1\geq1\), \(x_2\geq2\), \(x_3\geq3\), \(x_4\geq4\), what is the count?

Explanation opens after your attempt
Correct Answer

B. \(^{15}C_3\)

Explanation

Simple Explanation

Minimum sum (10) हटाने पर (12) बचता है और (4) variables में distribute होता है। परीक्षा में lower bounds subtract करके stars and bars लगाएं। / After removing the minimum sum (10), (12) remains and is distributed among (4) variables. In exams subtract lower bounds and use stars and bars.

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Question 569/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(x_1+x_2+x_3=19\) में \(0\leq x_i\leq7\) हो, तो valid count किस expression से मिलेगा?

If \(x_1+x_2+x_3=19\) and \(0\leq x_i\leq7\), which expression gives the valid count?

Explanation opens after your attempt
Correct Answer

A. \(^{21}C_2-3{}^{13}C_2+3{}^{5}C_2\)

Explanation

Simple Explanation

Upper violation \(x_i\geq8\) है और inclusion-exclusion applied होता है। परीक्षा में upper bound (7) के लिए shift (8) लें। / The upper violation is \(x_i\geq8\) and inclusion-exclusion is applied. In exams use shift (8) for upper bound (7).

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Question 570/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(x_1+x_2+x_3+x_4=16\) में exactly (1) variable zero हो और बाकी positive हों, तो count क्या है?

In \(x_1+x_2+x_3+x_4=16\), if exactly (1) variable is zero and the rest are positive, what is the count?

Explanation opens after your attempt
Correct Answer

A. \(^{4}C_1{}^{15}C_2\)

Explanation

Simple Explanation

Zero variable चुनें और बाकी (3) variables में positive sum (16) बांटें। परीक्षा में exactly zero cases को positive distribution में बदलें। / Choose the zero variable and split positive sum (16) among the remaining (3) variables. In exams convert exactly-zero cases into positive distribution.

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