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

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Question 171/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(^{n}C_r\) की व्युत्पत्ति में (r!(n-r)!) से भाग देने का grouped meaning क्या है?

What is the grouped meaning of dividing by (r!(n-r)!) in the derivation of \(^{n}C_r\)?

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Correct Answer

A. दो groups के अंदर के क्रम हटानाRemoving orders inside two groups

Explanation

Simple Explanation

(n!) में चुने और न चुने groups के internal orders अलग गिने जाते हैं। परीक्षा में group division में internal order हटाएँ। / In (n!), internal orders of chosen and unchosen groups are counted separately. In exams remove internal order in group division.

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Question 172/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि (n) अलग वस्तुओं को (r) और (n-r) के दो unlabelled groups में बाँटना हो तो formula कौन-सा है?

If (n) distinct objects are divided into two unlabelled groups of sizes (r) and (n-r), what is the formula?

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Correct Answer

B. \(^{n}C_r\)

Explanation

Simple Explanation

एक group चुनते ही दूसरा group तय हो जाता है इसलिए \(^{n}C_r\) पर्याप्त है। परीक्षा में complement fixed हो तो एक ही selection गिनें। / Once one group is chosen, the other group is fixed, so \(^{n}C_r\) is enough. In exams count only one selection when the complement is fixed.

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Question 173/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि (8) players में से (3) की team और फिर captain उसी team से चुनना हो तो expression क्या होगा?

If a team of (3) is selected from (8) players and then a captain is chosen from that team, what is the expression?

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Correct Answer

A. \(^{8}C_3\times3\)

Explanation

Simple Explanation

पहले team के \(^{8}C_3\) तरीके हैं और captain के (3) विकल्प हैं। परीक्षा में selection के बाद extra role हो तो multiply करें। / First there are \(^{8}C_3\) ways to form the team and (3) choices for captain. In exams multiply when an extra role follows selection.

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Question 174/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(^{8}C_3\times3\) को सीधे किस permutation relation से भी समझ सकते हैं?

By which permutation relation can \(^{8}C_3\times3\) also be understood directly?

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Correct Answer

C. \(^{8}P_3\div2!\)

Explanation

Simple Explanation

तीन में से एक captain distinguished है इसलिए बाकी (2) members का order हटाने पर \(^{8}P_3\div2!\) मिलता है। परीक्षा में roles की संख्या देखकर overcounting घटाएँ। / One of the three is distinguished as captain, so removing the order of the remaining (2) members gives \(^{8}P_3\div2!\). In exams reduce overcounting by checking the roles.

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Question 175/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि word में (9) letters हैं जिनमें (A) (2) बार, (B) (2) बार और (C) (3) बार आता है तो divisor क्या होगा?

If a word has (9) letters with (A) twice, (B) twice, and (C) three times, what will be the divisor?

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Correct Answer

A. (2!2!3!)

Explanation

Simple Explanation

हर repeated group के internal orders अलग नहीं होते इसलिए (2!2!3!) से भाग देते हैं। परीक्षा में repeated letters को अलग-अलग group मानें। / Internal orders of each repeated group are not different, so divide by (2!2!3!). In exams treat repeated letters as separate groups.

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Question 176/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(MISSISSIPPI) जैसे शब्दों में arrangements निकालते समय कौन-सा principle मुख्य है?

Which principle is main while finding arrangements of words like (MISSISSIPPI)?

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Correct Answer

A. Repeated letters के कारण division principleDivision principle due to repeated letters

Explanation

Simple Explanation

समान अक्षरों की अदला-बदली नया arrangement नहीं बनाती। परीक्षा में repeated letters की frequency लिखकर factorial से भाग दें। / Interchanging identical letters does not make a new arrangement. In exams write frequencies of repeated letters and divide by factorials.

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Question 177/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

(^{n}P_r=^{n}P_{r-1}(n-r+1)) में नया factor क्या दर्शाता है?

What does the new factor represent in (^{n}P_r=^{n}P_{r-1}(n-r+1))?

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Correct Answer

A. (r)वें स्थान के विकल्पChoices for the (r)th position

Explanation

Simple Explanation

पहले (r-1) स्थान भरने के बाद (r)वें स्थान पर (n-r+1) विकल्प बचते हैं। परीक्षा में recurrence को अगले स्थान के विचार से जोड़ें। / After filling the first (r-1) positions (n-r+1) choices remain for the (r)th position. In exams connect recurrence with the next-place idea.

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Question 178/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

गोल मेज पर (8) लोगों को बैठाने और line में बैठाने के counts का ratio कौन-सा है?

What is the ratio of seating (8) people around a round table to seating them in a line?

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Correct Answer

A. \(\frac{1}{8}\)

Explanation

Simple Explanation

Circular count (7!) और linear count (8!) है इसलिए ratio \(\frac{7!}{8!}=\frac{1}{8}\) है। परीक्षा में circular count में rotations हटते हैं। / The circular count is (7!) and the linear count is (8!), so the ratio is \(\frac{7!}{8!}=\frac{1}{8}\). In exams rotations are removed in circular counting.

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Question 179/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि (6) distinct beads की necklace में mirror images same माने जाएँ तो count किससे मिलेगा?

If mirror images are considered same in a necklace of (6) distinct beads, which gives the count?

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Correct Answer

B. \(\frac{5!}{2}\)

Explanation

Simple Explanation

पहले rotation हटाने से (5!) आता है फिर reflection same होने से (2) से भाग देते हैं। परीक्षा में necklace में rotation और reflection दोनों देखें। / Removing rotation first gives (5!), then reflection being same makes us divide by (2). In exams check both rotation and reflection in necklaces.

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Question 180/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \(^{n}C_r=^{n}C_{r+2}\) और दोनों निचले सूचक अलग हैं तो कौन-सा संबंध संभव है?

If \(^{n}C_r=^{n}C_{r+2}\) and the two lower indices are different, which relation is possible?

Explanation opens after your attempt
Correct Answer

A. (r+(r+2)=n)

Explanation

Simple Explanation

बराबर संचयों में निचले सूचक समान या पूरक होते हैं। यहाँ अलग सूचक हैं इसलिए उनका योग (n) होगा। / In equal combinations the lower indices are equal or complementary. Here the indices are different so their sum is (n).

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