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

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Question 471/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3+x_4=18\) में exactly (2) variables zero हों, तो count क्या है?

In \(x_1+x_2+x_3+x_4=18\), if exactly (2) variables are zero, what is the count?

Explanation opens after your attempt
Correct Answer

A. \(^{4}C_2{}^{17}C_1\)

Explanation

Simple Explanation

पहले zero variables चुनें, फिर बाकी दो variables positive sum (18) बनाते हैं। परीक्षा में exactly zero cases में positive distribution लगाएं। / Choose the zero variables first, then the remaining two variables form a positive sum of (18). In exams use positive distribution for exactly-zero cases.

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Question 472/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3=15\) में \(0\leq x_i\leq6\) हो, तो inclusion-exclusion में कौन-सा expression सही है?

In \(x_1+x_2+x_3=15\) with \(0\leq x_i\leq6\), which expression is correct by inclusion-exclusion?

Explanation opens after your attempt
Correct Answer

A. \(^{17}C_2-3{}^{10}C_2+3{}^{3}C_2\)

Explanation

Simple Explanation

Upper bound तोड़ने पर \(x_i\geq7\) के cases घटते और double violations जुड़ते हैं। परीक्षा में bounded solutions में (7) shift करें। / Cases with \(x_i\geq7\) are subtracted and double violations are added. In exams shift by (7) for bounded solutions.

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Question 473/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(x_1+x_2+x_3+x_4=10\) में हर \(x_i\leq4\) हो, तो valid count किस expression से मिलेगा?

If \(x_1+x_2+x_3+x_4=10\) and every \(x_i\leq4\), which expression gives the valid count?

Explanation opens after your attempt
Correct Answer

A. \(^{13}C_3-4{}^{8}C_3+6{}^{3}C_3\)

Explanation

Simple Explanation

Violation \(x_i\geq5\) से शुरू होती है और inclusion-exclusion लागू होता है। परीक्षा में upper limit (4) हो तो (5) subtract shift लें। / A violation begins with \(x_i\geq5\), so inclusion-exclusion applies. In exams use a subtract shift of (5) for upper limit (4).

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Question 474/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) letters की derangement संख्या \(D_n\) के लिए recurrence (D_n=(n-1)\(D_{n-1}+D_{n-2}\)) में (n-1) factor क्यों आता है?

In the derangement recurrence (D_n=(n-1)\(D_{n-1}+D_{n-2}\)), why does the factor (n-1) appear?

Explanation opens after your attempt
Correct Answer

A. पहले letter की गलत position चुनने के लिएTo choose the wrong position of the first letter

Explanation

Simple Explanation

पहला letter अपनी original position छोड़कर (n-1) positions में जा सकता है। परीक्षा में derangement recurrence में first object की गलत choice देखें। / The first letter can go to (n-1) positions other than its original position. In exams watch the first object's wrong choice in derangement recurrence.

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Question 475/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(D_n=n!\left(1-\frac{1}{1!}+\frac{1}{2!}-\cdots+(-1)^n\frac{1}{n!}\right)) किस principle से आता है?

The formula (D_n=n!\left(1-\frac{1}{1!}+\frac{1}{2!}-\cdots+(-1)^n\frac{1}{n!}\right)) comes from which principle?

Explanation opens after your attempt
Correct Answer

A. Inclusion-exclusion

Explanation

Simple Explanation

Fixed points वाले permutations को alternating तरीके से घटाया और जोड़ा जाता है। परीक्षा में कोई object सही स्थान पर न हो तो inclusion-exclusion सोचें। / Permutations with fixed points are subtracted and added alternately. In exams think of inclusion-exclusion when no object is in its correct place.

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Question 476/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(5) letters को envelopes में इस प्रकार रखना है कि कोई letter सही envelope में न जाए। Count कौन-सी है?

(5) letters are placed into envelopes so that no letter goes into its correct envelope. Which count is correct?

Explanation opens after your attempt
Correct Answer

A. \(D_5=44\)

Explanation

Simple Explanation

यह (5) objects का derangement है और \(D_5=44\) होता है। परीक्षा में letters-envelopes mismatch को derangement pattern मानें। / This is the derangement of (5) objects and \(D_5=44\). In exams treat letters-envelope mismatch as a derangement pattern.

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Question 477/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) people को (n) seats पर इस तरह बैठाना कि exactly (k) people सही seats पर हों, count क्या है?

What is the count for seating (n) people in (n) seats so that exactly (k) people are in their correct seats?

Explanation opens after your attempt
Correct Answer

A. \(^{n}C_kD_{n-k}\)

Explanation

Simple Explanation

पहले सही बैठे (k) people चुनें, फिर बाकी (n-k) का derangement करें। परीक्षा में exactly fixed points के लिए choose fixed plus derange rest करें। / First choose the (k) correctly seated people, then derange the remaining (n-k). In exams use choose fixed plus derange rest for exactly fixed points.

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Question 478/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct objects की circular arrangements में ((n-1)!) को (n!) से derive करने का सही कारण क्या है?

What is the correct reason for deriving ((n-1)!) from (n!) for circular arrangements of (n) distinct objects?

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Correct Answer

A. हर circular arrangement (n) rotations से linear arrangements में गिनी जाती हैEach circular arrangement is counted as (n) rotations in linear arrangements

Explanation

Simple Explanation

Linear (n!) count में rotations duplicate होते हैं। परीक्षा में circular arrangement में rotational overcount को (n) से divide करें। / Rotations are duplicates in the linear count (n!). In exams divide rotational overcount by (n) in circular arrangements.

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Question 479/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(n) distinct beads की bracelet arrangements में (\frac{(n-1)!}{2}) कब सही है?

When is (\frac{(n-1)!}{2}) correct for bracelet arrangements of (n) distinct beads?

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Correct Answer

A. जब rotations और reflections दोनों same माने जाएंWhen both rotations and reflections are considered the same

Explanation

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Bracelet में mirror images same मानी जाती हैं। परीक्षा में reflection same होने पर circular count को (2) से divide करें। / In a bracelet, mirror images are considered the same. In exams divide the circular count by (2) when reflection is the same.

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Question 480/600 Expert Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(8) people को round table पर बैठाना है और (A), (B), (C) साथ रहें। Count कौन-सी है?

(8) people are seated around a round table and (A), (B), (C) stay together. Which count is correct?

Explanation opens after your attempt
Correct Answer

A. \(5!\cdot3!\)

Explanation

Simple Explanation

तीन लोगों का one block और बाकी (5) people मिलकर (6) circular objects बनाते हैं, इसलिए (5!) arrangements हैं। परीक्षा में circular block count में objects minus one factorial लें। / The block of three people and the remaining (5) people form (6) circular objects, giving (5!) arrangements. In exams use one less factorial for circular block objects.

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