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

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Question 461/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि (n) अलग वस्तुओं को (a,b,c) आकार के labelled समूहों में बांटना हो और (a+b+c=n), तो सूत्र कौन-सा है?

If (n) distinct objects are divided into labelled groups of sizes (a,b,c) and (a+b+c=n), which formula is correct?

Explanation opens after your attempt
Correct Answer

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

Explanation

Simple Explanation

Labelled groups fixed हैं और हर group के अंदर order नहीं गिना जाता। परीक्षा में multinomial denominator में group-size factorials रखें। / Labelled groups are fixed and order inside each group is not counted. In exams put group-size factorials in the multinomial denominator.

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Question 462/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(12) अलग वस्तुओं को (4,4,4) के unlabelled समूहों में बांटने का formula क्या होगा?

What is the formula for dividing (12) distinct objects into unlabelled groups of (4,4,4)?

Explanation opens after your attempt
Correct Answer

B. \(\frac{12!}{4!4!4!3!}\)

Explanation

Simple Explanation

तीनों groups equal size और unlabelled हैं, इसलिए groups की (3!) अदला-बदली भी हटती है। परीक्षा में समान unlabelled groups पर extra factorial divide करें। / All three groups have equal size and are unlabelled, so the (3!) interchange of groups is also removed. In exams divide extra by factorial for equal unlabelled groups.

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Question 463/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(\frac{n!}{a!b!c!}\) को \({}^{n}C_a{}^{n-a}C_b\) से जोड़ने की शर्त क्या है?

What condition connects \(\frac{n!}{a!b!c!}\) with \({}^{n}C_a{}^{n-a}C_b\)?

Explanation opens after your attempt
Correct Answer

A. (a+b+c=n)

Explanation

Simple Explanation

पहले (a) चुनें, फिर बचे में से (b), और (c) अपने आप तय होता है। परीक्षा में sequential selection को factorial form से मिलाएं। / Choose (a) first, then (b) from the remaining, and (c) is automatically fixed. In exams match sequential selection with factorial form.

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Question 464/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct objects को (r) distinct boxes में onto तरीके से भेजने में inclusion-exclusion का सही सूत्र कौन-सा है?

Which inclusion-exclusion formula counts onto mappings from (n) distinct objects to (r) distinct boxes?

Explanation opens after your attempt
Correct Answer

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

Explanation

Simple Explanation

Total functions से empty boxes वाले cases inclusion-exclusion द्वारा हटते हैं। परीक्षा में onto का अर्थ हर box non-empty समझें। / Cases with empty boxes are removed from total functions by inclusion-exclusion. In exams read onto as every box being non-empty.

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Question 465/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct objects को (3) distinct boxes में onto distribute करने की संख्या क्या है?

What is the number of onto distributions of (n) distinct objects into (3) distinct boxes?

Explanation opens after your attempt
Correct Answer

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

Explanation

Simple Explanation

तीन empty choices घटती हैं, फिर दो empty boxes के over-subtraction को जोड़ा जाता है। परीक्षा में onto distribution में inclusion-exclusion सावधानी से लगाएं। / Three empty-box choices are subtracted, then over-subtraction of two empty boxes is added. In exams apply inclusion-exclusion carefully for onto distribution.

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Question 466/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct objects में से कोई object छोड़े बिना उन्हें (2) non-empty labelled boxes में बांटने की संख्या क्या है?

What is the number of ways to distribute (n) distinct objects into (2) non-empty labelled boxes without leaving any object?

Explanation opens after your attempt
Correct Answer

A. \(2^n-2\)

Explanation

Simple Explanation

Total \(2^n\) assignments में दोनों all-in-one empty-box cases invalid हैं। परीक्षा में labelled boxes हों तो empty cases अलग से घटाएं। / From total \(2^n\) assignments, the two all-in-one empty-box cases are invalid. In exams subtract empty cases separately for labelled boxes.

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Question 467/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct objects को (2) non-empty unlabelled groups में बांटने की संख्या कौन-सी है?

What is the number of ways to divide (n) distinct objects into (2) non-empty unlabelled groups?

Explanation opens after your attempt
Correct Answer

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

Explanation

Simple Explanation

Labelled count में दोनों groups की अदला-बदली दो बार गिनती है। परीक्षा में unlabelled दो groups के लिए (2) से divide करें। / In the labelled count, interchanging the two groups counts each division twice. In exams divide by (2) for two unlabelled groups.

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Question 468/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct objects से (r) objects repetition के साथ बिना order चुने जाएं, तो formula किससे आता है?

If (r) objects are chosen from (n) distinct objects with repetition and without order, which idea gives the formula?

Explanation opens after your attempt
Correct Answer

A. Stars and bars

Explanation

Simple Explanation

यह multiset selection है और answer \({}^{n+r-1}C_r\) होता है। परीक्षा में repetition with no order को bars method से करें। / This is multiset selection and the answer is \({}^{n+r-1}C_r\). In exams handle repetition with no order by the bars method.

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Question 469/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3+x_4=25\) और \(x_i\geq2\) हो, तो solutions की संख्या कौन-सी है?

If \(x_1+x_2+x_3+x_4=25\) and \(x_i\geq2\), what is the number of solutions?

Explanation opens after your attempt
Correct Answer

B. \(^{20}C_3\)

Explanation

Simple Explanation

चार variables को पहले (2) देने पर (17) बचता है। परीक्षा में lower bound घटाकर non-negative stars and bars लगाएं। / Give (2) first to the four variables, leaving (17). In exams subtract the lower bound and apply non-negative stars and bars.

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Question 470/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3=20\) में \(x_1\geq3\), \(x_2\geq4\), \(x_3\geq5\) हो, तो count क्या है?

In \(x_1+x_2+x_3=20\), if \(x_1\geq3\), \(x_2\geq4\), \(x_3\geq5\), what is the count?

Explanation opens after your attempt
Correct Answer

A. \(^{10}C_2\)

Explanation

Simple Explanation

Minimum (3+4+5=12) हटाने पर (8) बचता है। परीक्षा में unequal lower bounds को पहले shift करें। / Removing the minimum (3+4+5=12) leaves (8). In exams shift unequal lower bounds first.

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