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

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Question 511/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) distinct objects को (4) labelled boxes में onto भेजने की संख्या कौन-सी है?

What is the number of onto distributions of (n) distinct objects into (4) labelled boxes?

Explanation opens after your attempt
Correct Answer

A. \(4^n-4\cdot3^n+6\cdot2^n-4\)

Explanation

Simple Explanation

Empty boxes को inclusion-exclusion से घटाया और जोड़ा जाता है। परीक्षा में onto का मतलब हर labelled box non-empty समझें। / Empty boxes are subtracted and added by inclusion-exclusion. In exams interpret onto as every labelled box being non-empty.

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Question 512/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) distinct objects को (3) non-empty unlabelled groups में बांटने की संख्या किससे जुड़ती है?

The number of ways to divide (n) distinct objects into (3) non-empty unlabelled groups is connected with which form?

Explanation opens after your attempt
Correct Answer

A. (\frac{1}{3!}\left\(3^n-3\cdot2^n+3\right\))

Explanation

Simple Explanation

पहले (3) labelled non-empty groups गिनते हैं, फिर labels की (3!) अदला-बदली हटाते हैं। परीक्षा में unlabelled groups के लिए labelled count divide करें। / First count (3) labelled non-empty groups, then remove the (3!) label permutations. In exams divide the labelled count for unlabelled groups.

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Question 513/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3+x_4=30\) में \(x_1\geq2\), \(x_2\geq3\), \(x_3\geq4\), \(x_4\geq5\) हो, तो count क्या है?

In \(x_1+x_2+x_3+x_4=30\), if \(x_1\geq2\), \(x_2\geq3\), \(x_3\geq4\), \(x_4\geq5\), what is the count?

Explanation opens after your attempt
Correct Answer

A. \(^{19}C_3\)

Explanation

Simple Explanation

Minimum sum (14) हटाने पर (16) बचता है, इसलिए \({}^{16+4-1}C_{3}\) मिलता है। परीक्षा में unequal lower bounds पहले subtract करें। / After removing the minimum sum (14), (16) remains, so \({}^{16+4-1}C_{3}\) is obtained. In exams subtract unequal lower bounds first.

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Question 514/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3=24\) में \(0\leq x_i\leq8\) हो, तो valid count किस expression से मिलेगा?

If \(x_1+x_2+x_3=24\) and \(0\leq x_i\leq8\), which expression gives the valid count?

Explanation opens after your attempt
Correct Answer

B. (1)

Explanation

Simple Explanation

कुल (24) और तीन variables की maximum (8) होने से केवल ((8,8,8)) संभव है। परीक्षा में inclusion-exclusion से पहले extreme feasibility देखें। / Since the total is (24) and the maximum of each of the three variables is (8), only ((8,8,8)) is possible. In exams check extreme feasibility before inclusion-exclusion.

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Question 515/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3+x_4=17\) में \(0\leq x_i\leq5\) हो, तो valid count कौन-सा है?

If \(x_1+x_2+x_3+x_4=17\) and \(0\leq x_i\leq5\), what is the valid count?

Explanation opens after your attempt
Correct Answer

A. \(^{20}C_3-4{}^{14}C_3+6{}^{8}C_3-4{}^{2}C_3\)

Explanation

Simple Explanation

Violation \(x_i\geq6\) से शुरू होती है और inclusion-exclusion लागू होता है। परीक्षा में upper bound (5) हो तो shift (6) लें। / A violation starts at \(x_i\geq6\), so inclusion-exclusion applies. In exams use shift (6) for upper bound (5).

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Question 516/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3+x_4+x_5=12\) में exactly (3) variables positive हों, तो count क्या होगा?

In \(x_1+x_2+x_3+x_4+x_5=12\), if exactly (3) variables are positive, what is the count?

Explanation opens after your attempt
Correct Answer

A. \(^{5}C_3{}^{11}C_2\)

Explanation

Simple Explanation

पहले (3) positive variables चुनें, फिर (12) को (3) positive parts में बांटें। परीक्षा में exactly positive variables में choose variables plus positive stars-bars करें। / Choose the (3) positive variables first, then split (12) into (3) positive parts. In exams use choose variables plus positive stars and bars for exactly positive variables.

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Question 517/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(20) identical balls को (5) boxes में बांटना है और exactly (2) boxes empty हों। सही count कौन-सी है?

(20) identical balls are distributed into (5) boxes and exactly (2) boxes are empty. Which count is correct?

Explanation opens after your attempt
Correct Answer

A. \(^{5}C_2{}^{19}C_2\)

Explanation

Simple Explanation

पहले empty boxes चुनें, फिर बाकी (3) boxes में positive distribution करें। परीक्षा में exactly empty को boxes selection और positive distribution में तोड़ें। / First choose the empty boxes, then distribute positively into the remaining (3) boxes. In exams split exactly empty into box selection and positive distribution.

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Question 518/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(D_n\) के लिए (D_n=nD_{n-1}+(-1)^n) किस formula से निकलता है?

From which formula does (D_n=nD_{n-1}+(-1)^n) arise for \(D_n\)?

Explanation opens after your attempt
Correct Answer

A. Derangement inclusion-exclusion formula

Explanation

Simple Explanation

Derangement के alternating factorial expression को compare करने से यह recurrence मिलता है। परीक्षा में derangement recurrence के लिए inclusion-exclusion form याद रखें। / Comparing the alternating factorial expression for derangements gives this recurrence. In exams remember the inclusion-exclusion form for derangement recurrence.

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Question 519/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

Exactly (2) fixed points वाले permutations of (7) objects की संख्या क्या है?

What is the number of permutations of (7) objects with exactly (2) fixed points?

Explanation opens after your attempt
Correct Answer

A. \(^{7}C_2D_5\)

Explanation

Simple Explanation

पहले सही रहने वाले (2) objects चुनें और बाकी (5) objects derange करें। परीक्षा में exactly fixed points के लिए choose fixed plus derange rest लगाएं। / First choose the (2) objects that stay fixed and derange the remaining (5). In exams use choose fixed plus derange rest for exactly fixed points.

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Question 520/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(6) letters और (6) envelopes में कोई letter सही envelope में न जाए, तो count कौन-सी है?

For (6) letters and (6) envelopes, if no letter goes into its correct envelope, which count is correct?

Explanation opens after your attempt
Correct Answer

A. \(D_6=265\)

Explanation

Simple Explanation

यह (6) objects का derangement है और \(D_6=265\) होता है। परीक्षा में letters-envelope mismatch को derangement समझें। / This is a derangement of (6) objects and \(D_6=265\). In exams treat letter-envelope mismatch as derangement.

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