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

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Question 291/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{20}C_a=^{20}C_{a+6}\) और indices अलग हैं, तो (a) क्या होगा?

If \(^{20}C_a=^{20}C_{a+6}\) and the indices are different, what is (a)?

Explanation opens after your attempt
Correct Answer

B. (7)

Explanation

Simple Explanation

पूरक indices का योग (20) होगा इसलिए (a+a+6=20) और (a=7) है। परीक्षा में equal combinations से linear equation बनाएं। / Complementary indices have sum (20), so (a+a+6=20) and (a=7). In exams form a linear equation from equal combinations.

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Question 292/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{21}C_x=^{21}C_{2x}\) और \(x\neq2x\) है, तो (x) क्या होगा?

If \(^{21}C_x=^{21}C_{2x}\) and \(x\neq2x\), what is (x)?

Explanation opens after your attempt
Correct Answer

C. (7)

Explanation

Simple Explanation

अलग lower indices पूरक हैं इसलिए (x+2x=21) और (x=7) है। परीक्षा में equal (C) terms को complement rule से हल करें। / Different lower indices are complementary, so (x+2x=21) and (x=7). In exams solve equal (C) terms using the complement rule.

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Question 293/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}C_1+^{n}C_2+\cdots+^{n}C_n=127\) हो, तो (n) कितना होगा?

If \(^{n}C_1+^{n}C_2+\cdots+^{n}C_n=127\), what is (n)?

Explanation opens after your attempt
Correct Answer

B. (7)

Explanation

Simple Explanation

यह non-empty selections का sum \(2^n-1\) है और \(2^7-1=127\) है। परीक्षा में empty selection हटाने पर (1) घटाएँ। / This is the sum of non-empty selections \(2^n-1\), and \(2^7-1=127\). In exams subtract (1) when the empty selection is removed.

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Question 294/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}C_0+^{n}C_1+\cdots+^{n}C_n=512\) हो, तो (n) क्या होगा?

If \(^{n}C_0+^{n}C_1+\cdots+^{n}C_n=512\), what is (n)?

Explanation opens after your attempt
Correct Answer

C. (9)

Explanation

Simple Explanation

कुल sum \(2^n\) है और \(2^9=512\) होता है। परीक्षा में all combinations sum को power of (2) से जोड़ें। / The total sum is \(2^n\), and \(2^9=512\). In exams connect the sum of all combinations with a power of (2).

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Question 295/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}C_0+^{n}C_2+^{n}C_4+\cdots=256\) हो, तो (n) क्या होगा?

If \(^{n}C_0+^{n}C_2+^{n}C_4+\cdots=256\), what is (n)?

Explanation opens after your attempt
Correct Answer

B. (9)

Explanation

Simple Explanation

Even indexed sum \(2^{n-1}\) है और \(2^8=256\) से (n=9) है। परीक्षा में even और odd sums बराबर याद रखें। / The even indexed sum is \(2^{n-1}\), and \(2^8=256\) gives (n=9). In exams remember that even and odd sums are equal.

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Question 296/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

\(^{n}C_r\) की factorial derivation में (r!) और ((n-r)!) दोनों क्यों आते हैं?

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

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Correct Answer

A. चुने और न चुने groups के internal orders हटाने के लिएTo remove internal orders of chosen and unchosen groups

Explanation

Simple Explanation

(n!) वाली गिनती में दोनों groups के अंदर के क्रम extra गिने जाते हैं। परीक्षा में (r!) और ((n-r)!) दोनों corrections समझें। / In the (n!) count, the orders inside both groups are counted extra. In exams understand both corrections (r!) and ((n-r)!).

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Question 297/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि (n) objects को (3) labelled groups में sizes (a), (b), (c) में बाँटना हो, तो formula कौन-सा है?

If (n) objects are divided into (3) labelled groups of sizes (a), (b), (c), which formula is correct?

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Correct Answer

A. \(\frac{n!}{a!b!c!}\)

Explanation

Simple Explanation

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

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Question 298/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि (10) objects को (5) और (5) के दो unlabelled groups में बाँटना हो, तो expression क्या होगा?

If (10) objects are divided into two unlabelled groups of (5) and (5), what is the expression?

Explanation opens after your attempt
Correct Answer

A. \(\frac{^{10}C_5}{2!}\)

Explanation

Simple Explanation

पहला (5)-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 (5)-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 299/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

यदि \(^{n}P_4=360\times{}^{n}C_4\) कहा जाए, तो यह कथन कैसा है?

If it is said that \(^{n}P_4=360\times{}^{n}C_4\), what type of statement is it?

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Correct Answer

A. गलत, क्योंकि सही गुणक (4!=24) हैFalse, because the correct multiplier is (4!=24)

Explanation

Simple Explanation

\(^{n}P_r=^{n}C_r r!\) है, इसलिए (r=4) पर गुणक (24) होगा। परीक्षा में multiplier को factorial से verify करें। / \(^{n}P_r=^{n}C_r r!\), so for (r=4) the multiplier is (24). In exams verify the multiplier using factorials.

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Question 300/600 Medium Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 66

कौन-सा कथन \(^{n}C_r\), \(^{n}P_r\) और (r!) के संबंध को सबसे सही बताता है?

Which statement best describes the relation among \(^{n}C_r\), \(^{n}P_r\), and (r!)?

Explanation opens after your attempt
Correct Answer

A. \(^{n}P_r\) में चयन के साथ चुनी वस्तुओं की व्यवस्था भी गिनी जाती है\(^{n}P_r\) counts selection along with arrangement of selected objects

Explanation

Simple Explanation

Permutation combination से (r!) गुना होता है क्योंकि चुनी वस्तुओं को क्रम देना पड़ता है। परीक्षा में \(P=C\times r!\) को concept से याद रखें। / Permutation is (r!) times combination because the selected objects must be ordered. In exams remember \(P=C\times r!\) conceptually.

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