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

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Question 361/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(10) seats की row में (4) selected students को बैठाने और बाकी seats empty छोड़ने की count किस expression से जुड़ती है?

The count for seating (4) selected students in a row of (10) seats while leaving other seats empty is connected with which expression?

Explanation opens after your attempt
Correct Answer

B. \(^{10}P_4\)

Explanation

Simple Explanation

पहले seats चुनना और फिर students arrange करना \(^{10}C_4\cdot4!=^{10}P_4\) है। परीक्षा में seat-position और arrangement दोनों गिनें। / Choosing seats first and then arranging students gives \(^{10}C_4\cdot4!=^{10}P_4\). In exams count both positions and arrangements.

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Question 362/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(7) people में से (3) को क्रम वाली queue में चुनने की count \(^{7}C_3\cdot3!\) क्यों है?

Why is the count for choosing (3) people from (7) into an ordered queue equal to \(^{7}C_3\cdot3!\)?

Explanation opens after your attempt
Correct Answer

A. पहले group चुनते हैं फिर उस group को order देते हैंFirst choose the group then order that group

Explanation

Simple Explanation

Queue में selected people की order meaningful होती है। परीक्षा में ordered selection को combination times factorial से derive करें। / Order of selected people is meaningful in a queue. In exams derive ordered selection as combination times factorial.

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Question 363/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(8) अलग objects को (3), (3), और (2) के labelled groups में बांटने का formula कौन-सा है?

What is the formula for dividing (8) distinct objects into labelled groups of (3), (3), and (2)?

Explanation opens after your attempt
Correct Answer

A. \(\frac{8!}{3!3!2!}\)

Explanation

Simple Explanation

Labelled groups में group names fixed होते हैं इसलिए सिर्फ internal order हटता है। परीक्षा में fixed group sizes के factorial denominator लगाएं। / For labelled groups, group names are fixed so only internal order is removed. In exams use factorial denominators for fixed group sizes.

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Question 364/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(8) अलग objects को sizes (3), (3), और (2) के unlabelled groups में बांटते समय अतिरिक्त division किससे होगा?

When (8) distinct objects are divided into unlabelled groups of sizes (3), (3), and (2), by what extra factor do we divide?

Explanation opens after your attempt
Correct Answer

A. (2!)

Explanation

Simple Explanation

दो groups का size (3) समान है इसलिए उन groups की अदला-बदली same distribution देती है। परीक्षा में equal-sized unlabelled groups के factorial से extra divide करें। / Two groups have equal size (3), so interchanging those groups gives the same distribution. In exams divide extra by factorial of equal-sized unlabelled groups.

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Question 365/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

किस formula से (n) distinct objects को (r) distinct boxes में distribute किया जाता है जब हर object किसी एक box में जा सकता है?

Which formula distributes (n) distinct objects into (r) distinct boxes when each object can go into one box?

Explanation opens after your attempt
Correct Answer

C. \(r^n\)

Explanation

Simple Explanation

हर distinct object के लिए (r) independent choices हैं। परीक्षा में distinct objects और distinct boxes में power rule लगाएं। / Each distinct object has (r) independent choices. In exams use the power rule for distinct objects and distinct boxes.

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Question 366/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) distinct objects को (r) distinct boxes में onto distribution गिनने के लिए inclusion-exclusion का रूप कौन-सा है?

Which inclusion-exclusion form counts onto distributions of (n) distinct objects into (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

Empty boxes को exclude करने के लिए inclusion-exclusion used होता है। परीक्षा में onto शब्द दिखे तो खाली boxes घटाने की सोचें। / Inclusion-exclusion is used to exclude empty boxes. In exams think of removing empty boxes when the word onto appears.

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Question 367/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) लोगों को (r) distinct rooms में रखना है और rooms empty हो सकते हैं। Count \(r^n\) क्यों है?

(n) people are to be placed into (r) distinct rooms and rooms may be empty. Why is the count \(r^n\)?

Explanation opens after your attempt
Correct Answer

A. क्योंकि हर person के लिए (r) room choices independent हैंBecause each person has (r) independent room choices

Explanation

Simple Explanation

Distinct persons के independent choices multiply होते हैं। परीक्षा में व्यक्ति distinct हों तो balls-and-boxes को power से जोड़ें। / Independent choices of distinct people multiply. In exams connect distinct-person box problems with powers.

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Question 368/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) identical coins को (r) बच्चों में बांटने में कौन-सी condition formula \({}^{n+r-1}C_{n}\) देती है?

Which condition gives the formula \({}^{n+r-1}C_{n}\) for distributing (n) identical coins among (r) children?

Explanation opens after your attempt
Correct Answer

B. किसी बच्चे को zero coin मिल सकता हैA child may get zero coins

Explanation

Simple Explanation

Zero allowed हो तो (n) stars और (r-1) bars रखे जाते हैं। परीक्षा में \({}^{n+r-1}C_n\) और \({}^{n+r-1}C_{r-1}\) same मानें। / When zero is allowed, place (n) stars and (r-1) bars. In exams treat \({}^{n+r-1}C_n\) and \({}^{n+r-1}C_{r-1}\) as the same.

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Question 369/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3=12\) के non-negative integer solutions की count कौन-सी है?

What is the count of non-negative integer solutions of \(x_1+x_2+x_3=12\)?

Explanation opens after your attempt
Correct Answer

B. \({}^{14}C_2\)

Explanation

Simple Explanation

Non-negative solutions stars and bars से \({}^{12+3-1}C_{3-1}\) होते हैं। परीक्षा में equation solutions को distribution problem बनाएं। / Non-negative solutions by stars and bars are \({}^{12+3-1}C_{3-1}\). In exams convert equation solutions into distribution problems.

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Question 370/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(x_1+x_2+x_3+x_4=15\) में सभी \(x_i\geq1\) हों तो count कौन-सी है?

If all \(x_i\geq1\) in \(x_1+x_2+x_3+x_4=15\), what is the count?

Explanation opens after your attempt
Correct Answer

C. \({}^{11}C_3\)

Explanation

Simple Explanation

पहले हर variable को (1) दें, फिर (11) बचते हैं। परीक्षा में positive solutions में total से variables की संख्या घटाएं। / Give (1) to each variable first, then (11) remain. In exams subtract the number of variables for positive solutions.

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