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

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Derivations of formulas and their connections Practice Questions

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Question 191/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \(\frac{^{n}C_r}{^{n}C_{r-1}}=1\) हो तो कौन-सा संबंध सही है?

If \(\frac{^{n}C_r}{^{n}C_{r-1}}=1\), which relation is correct?

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Correct Answer

A. (n-r+1=r)

Explanation

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अनुपात \(\frac{n-r+1}{r}\) है और इसे (1) रखने पर (n-r+1=r) मिलता है। परीक्षा में ratio identity से equation बनाइए। / The ratio is \(\frac{n-r+1}{r}\), and setting it equal to (1) gives (n-r+1=r). In exams form an equation using the ratio identity.

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Question 192/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि \(^{n}P_r=^{n}P_{r-1}\times5\) हो तो (5) किस expression के बराबर है?

If \(^{n}P_r=^{n}P_{r-1}\times5\), then (5) is equal to which expression?

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Correct Answer

A. (n-r+1)

Explanation

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recurrence में extra factor (r)वें स्थान के choices यानी (n-r+1) होता है। परीक्षा में multiplier को next-place choices मानें। / In the recurrence, the extra factor is the choices for the (r)th place, which is (n-r+1). In exams treat the multiplier as next-place choices.

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Question 193/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(^{n}C_r\) के formula से \(^{n}C_{n-r}\) निकालने पर denominator में कौन-से factorial आते हैं?

Using the formula for \(^{n}C_r\), which factorials appear in the denominator of \(^{n}C_{n-r}\)?

Explanation opens after your attempt
Correct Answer

A. (r!(n-r)!)

Explanation

Simple Explanation

(^{n}C_{n-r}=\frac{n!}{(n-r)!r!}) है जो वही denominator देता है। परीक्षा में symmetry को factorial से verify करें। / (^{n}C_{n-r}=\frac{n!}{(n-r)!r!}), which gives the same denominator. In exams verify symmetry using factorials.

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Question 194/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(^{n}C_r\) में (r) और (n-r) को interchange करने से value क्यों नहीं बदलती?

Why does the value not change when (r) and (n-r) are interchanged in \(^{n}C_r\)?

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Correct Answer

A. Denominator में वही दो factorial रहते हैंThe same two factorials remain in the denominator

Explanation

Simple Explanation

(\frac{n!}{r!(n-r)!}) में (r!) और ((n-r)!) का क्रम बदलने से मान नहीं बदलता। परीक्षा में multiplication denominator में order matter नहीं करता। / In (\frac{n!}{r!(n-r)!}), changing the order of (r!) and ((n-r)!) does not change the value. In exams order does not matter in a product denominator.

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Question 195/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि (11) objects को (4) selected और (7) rejected में बाँटा जाए तो selected group count कौन-सा है?

If (11) objects are divided into (4) selected and (7) rejected objects, what is the selected group count?

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Correct Answer

A. \(^{11}C_4\)

Explanation

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(4) selected चुनते ही (7) rejected तय हो जाते हैं। परीक्षा में selected-rejected split को combination से जोड़ें। / Once (4) selected objects are chosen, the (7) rejected objects are fixed. In exams connect selected-rejected split with combination.

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Question 196/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(^{n}P_r\) के factorial formula में denominator ((n-r)!) किस वस्तु-समूह से जुड़ा है?

In the factorial formula of \(^{n}P_r\), the denominator ((n-r)!) is connected with which group of objects?

Explanation opens after your attempt
Correct Answer

A. न चुनी गई वस्तुएँUnchosen objects

Explanation

Simple Explanation

(n!) में न चुनी गई ((n-r)) वस्तुओं के अंदरूनी क्रम extra हैं इसलिए उन्हें हटाया जाता है। परीक्षा में permutation formula में remaining part पहचानें। / In (n!), internal orders of the unchosen ((n-r)) objects are extra, so they are removed. In exams identify the remaining part in the permutation formula.

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Question 197/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि (r=1) हो तो \(^{n}P_r=^{n}C_r\) क्यों होता है?

If (r=1), why is \(^{n}P_r=^{n}C_r\)?

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Correct Answer

A. क्योंकि (1!=1)Because (1!=1)

Explanation

Simple Explanation

एक चुनी वस्तु का केवल (1) order होता है। परीक्षा में (r=1) पर permutation और combination दोनों (n) होते हैं। / One selected object has only (1) order. In exams permutation and combination both equal (n) when (r=1).

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Question 198/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

यदि (r=0) हो तो \(^{n}P_r=^{n}C_r\) किस कारण होता है?

If (r=0), why is \(^{n}P_r=^{n}C_r\)?

Explanation opens after your attempt
Correct Answer

A. क्योंकि खाली चयन और खाली व्यवस्था दोनों (1) हैंBecause empty selection and empty arrangement are both (1)

Explanation

Simple Explanation

शून्य वस्तुओं के लिए count (1) माना जाता है। परीक्षा में (0!=1) और empty case याद रखें। / For zero objects the count is taken as (1). In exams remember (0!=1) and the empty case.

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Question 199/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

किस formula connection से \(^{10}P_3\div3!\) को सीधे \(^{10}C_3\) कहा जा सकता है?

By which formula connection can \(^{10}P_3\div3!\) be directly called \(^{10}C_3\)?

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Correct Answer

A. \(^{n}C_r=\frac{^{n}P_r}{r!}\)

Explanation

Simple Explanation

Combination permutation में गिने गए (r!) internal orders को हटाकर मिलता है। परीक्षा में ordered से unordered जाने पर divide करें। / Combination is obtained by removing (r!) internal orders counted in permutation. In exams divide when going from ordered to unordered.

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Question 200/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 64

\(^{n}C_r\) में (n-r) का व्यावहारिक अर्थ क्या है?

What is the practical meaning of (n-r) in \(^{n}C_r\)?

Explanation opens after your attempt
Correct Answer

B. न चुनी गई वस्तुओं की संख्याNumber of unchosen objects

Explanation

Simple Explanation

(r) वस्तुएँ चुनने के बाद (n-r) वस्तुएँ बचती हैं। परीक्षा में complement counting के लिए यह अर्थ याद रखें। / After choosing (r) objects (n-r) objects remain. In exams remember this meaning for complement counting.

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