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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 241/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \(^{18}C_{x}=^{18}C_{2x}\) और \(x\neq2x\) है तो (x) क्या होगा?

If \(^{18}C_x=^{18}C_{2x}\) and \(x\neq2x\), what is (x)?

Explanation opens after your attempt
Correct Answer

C. (6)

Explanation

Simple Explanation

अलग lower indices पूरक हैं इसलिए (x+2x=18) और (x=6) है। परीक्षा में equal (C) terms को complement rule से हल करें। / Different lower indices are complementary, so (x+2x=18) and (x=6). In exams solve equal (C) terms using the complement rule.

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Question 242/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \(^{n}C_1+^{n}C_2+\cdots+^{n}C_n=63\) हो तो (n) कितना होगा?

If \(^{n}C_1+^{n}C_2+\cdots+^{n}C_n=63\), what is (n)?

Explanation opens after your attempt
Correct Answer

B. (6)

Explanation

Simple Explanation

यह non-empty selections का sum \(2^n-1\) है और \(2^6-1=63\) है। परीक्षा में empty selection हटाने पर (1) घटाएँ। / This is the sum of non-empty selections \(2^n-1\), and \(2^6-1=63\). In exams subtract (1) when the empty selection is removed.

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Question 243/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \(^{n}C_0+^{n}C_1+\cdots+^{n}C_n=256\) हो तो (n) क्या होगा?

If \(^{n}C_0+^{n}C_1+\cdots+^{n}C_n=256\), what is (n)?

Explanation opens after your attempt
Correct Answer

C. (8)

Explanation

Simple Explanation

कुल sum \(2^n\) है और \(2^8=256\) होता है। परीक्षा में all combinations sum को power of (2) से जोड़ें। / The total sum is \(2^n\), and \(2^8=256\). In exams connect the sum of all combinations with a power of (2).

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Question 244/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \(^{n}C_0+^{n}C_2+^{n}C_4+\cdots=128\) हो तो (n) क्या होगा?

If \(^{n}C_0+^{n}C_2+^{n}C_4+\cdots=128\), what is (n)?

Explanation opens after your attempt
Correct Answer

B. (8)

Explanation

Simple Explanation

Even indexed sum \(2^{n-1}\) है और \(2^7=128\) से (n=8) है। परीक्षा में even और odd sums बराबर याद रखें। / The even indexed sum is \(2^{n-1}\), and \(2^7=128\) gives (n=8). In exams remember that even and odd sums are equal.

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Question 245/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(^{n}C_r\) की factorial derivation में ((n-r)!) क्यों आता है?

Why does ((n-r)!) appear in the factorial derivation of \(^{n}C_r\)?

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Correct Answer

A. न चुनी गई वस्तुओं के internal orders हटाने के लिएTo remove internal orders of unchosen objects

Explanation

Simple Explanation

(n!) वाली गिनती में न चुने गए समूह के क्रम भी extra गिने जाते हैं। परीक्षा में (r!) और ((n-r)!) दोनों corrections समझें। / In the (n!) count the orders inside the unchosen group are also counted extra. In exams understand both corrections (r!) and ((n-r)!).

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Question 246/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(^{n}P_r\) की factorial derivation में ((n-r)!) से भाग देने का अर्थ क्या है?

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

Explanation opens after your attempt
Correct Answer

A. बची हुई वस्तुओं के क्रम को ignore करनाIgnoring the order of remaining objects

Explanation

Simple Explanation

Permutation में केवल चुनी (r) वस्तुओं का order चाहिए और बची वस्तुओं का order नहीं चाहिए। परीक्षा में (n!) से unwanted tail हटाएँ। / Permutation needs the order of only the selected (r) objects and not the order of remaining objects. In exams remove the unwanted tail from (n!).

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Question 247/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि (n) objects को (3) labelled groups में sizes (a), (b), (c) में बाँटना हो तो denominator का रूप क्या होगा?

If (n) objects are divided into (3) labelled groups of sizes (a), (b), (c), what is the form of the denominator?

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Correct Answer

A. (a!b!c!)

Explanation

Simple Explanation

हर group के अंदर order महत्वहीन है इसलिए (a!b!c!) से भाग दिया जाता है। परीक्षा में labelled groups में groups को आपस में divide न करें। / Order inside each group is irrelevant, so divide by (a!b!c!). In exams do not divide among labelled groups themselves.

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Question 248/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि (n) objects को (2) equal unlabelled groups में बाँटना हो तो \(^{n}C_{n/2}\) के बाद किससे भाग देना होगा?

If (n) objects are divided into (2) equal unlabelled groups, by what should \(^{n}C_{n/2}\) be divided?

Explanation opens after your attempt
Correct Answer

A. (2!)

Explanation

Simple Explanation

दो equal groups unlabelled हैं इसलिए group interchange दो बार गिनता है। परीक्षा में equal unlabelled groups पर extra (2!) correction लगाएँ। / The two equal groups are unlabelled, so interchanging the groups counts twice. In exams apply the extra (2!) correction for equal unlabelled groups.

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Question 249/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि (8) objects को (4) और (4) के दो unlabelled groups में बाँटना हो तो expression क्या होगा?

If (8) objects are divided into two unlabelled groups of (4) and (4), what is the expression?

Explanation opens after your attempt
Correct Answer

A. \(\frac{^{8}C_4}{2!}\)

Explanation

Simple Explanation

पहला (4)-group चुनने पर दूसरा तय है और दोनों groups interchange होने से duplicate count आता है। परीक्षा में equal groups हों तो (2!) से भाग दें। / After choosing the first (4)-group the second is fixed, and interchanging the two groups creates duplicate count. In exams divide by (2!) for equal groups.

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Question 250/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(^{n}C_r\) और \(^{n}P_r\) के connection में कौन-सा कथन सबसे सही है?

Which statement is most correct about the connection between \(^{n}C_r\) and \(^{n}P_r\)?

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Correct Answer

A. \(^{n}C_r\) selection गिनता है और \(^{n}P_r\) ordered selection गिनता है\(^{n}C_r\) counts selection and \(^{n}P_r\) counts ordered selection

Explanation

Simple Explanation

Permutation में combination के साथ चुनी वस्तुओं की व्यवस्था भी जुड़ती है। परीक्षा में \(P=C\times r!\) को concept से याद रखें। / Permutation adds the arrangement of chosen objects to combination. In exams remember \(P=C\times r!\) conceptually.

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