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

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Question 351/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

((1+x)^n) में \(x^r\) का coefficient \(^{n}C_r\) क्यों होता है?

Why is the coefficient of \(x^r\) in ((1+x)^n) equal to \(^{n}C_r\)?

Explanation opens after your attempt
Correct Answer

A. क्योंकि (r) brackets से (x) और बाकी से (1) चुनते हैंBecause (x) is chosen from (r) brackets and (1) from the rest

Explanation

Simple Explanation

\(x^r\) बनाने के लिए (n) brackets में से (r) brackets चुनते हैं। परीक्षा में coefficient को selection से जोड़ें। / To form \(x^r\), choose (r) brackets from (n) brackets. In exams connect coefficients with selection.

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Question 352/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\(\sum_{r=0}^{n} {}^{n}C_r=2^n\) किस combinatorial idea से निकला है?

The identity \(\sum_{r=0}^{n} {}^{n}C_r=2^n\) is derived from which combinatorial idea?

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Correct Answer

B. हर वस्तु को चुनना या न चुननाSelecting or not selecting each object

Explanation

Simple Explanation

Left side all possible selection sizes को जोड़ता है और right side हर object के two choices देता है। परीक्षा में subset counting से यह identity याद रखें। / The left side adds all selection sizes and the right side gives two choices for each object. In exams remember this identity through subset counting.

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Question 353/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(\sum_{r=0}^{n} (-1)^r{}^{n}C_r=0) किस substitution से जुड़ा है?

The identity (\sum_{r=0}^{n} (-1)^r{}^{n}C_r=0) is connected with which substitution?

Explanation opens after your attempt
Correct Answer

C. ((1+x)^n) में (x=-1)(x=-1) in ((1+x)^n)

Explanation

Simple Explanation

(x=-1) रखने पर ((1-1)^n=0) मिलता है। परीक्षा में alternating binomial sum में (x=-1) सोचें। / Putting (x=-1) gives ((1-1)^n=0). In exams think of (x=-1) for alternating binomial sums.

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Question 354/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\({}^{m+n}C_r=\sum_{k=0}^{r}{}^{m}C_k{}^{n}C_{r-k}\) किस counting split से सिद्ध होता है?

The identity \({}^{m+n}C_r=\sum_{k=0}^{r}{}^{m}C_k{}^{n}C_{r-k}\) is proved by which counting split?

Explanation opens after your attempt
Correct Answer

B. दो groups से कुल (r) चुनने में (k) पहले group से लेनाTaking (k) from the first group while selecting total (r) from two groups

Explanation

Simple Explanation

कुल (r) selection में first group का count (k) बदलता है। परीक्षा में two-source selection को Vandermonde identity से जोड़ें। / In total (r) selections, the count from the first group varies as (k). In exams connect two-source selection with Vandermonde identity.

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Question 355/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\({}^{n}C_r=\frac{n}{r}{}^{n-1}C_{r-1}\) की counting व्याख्या क्या है?

What is the counting interpretation of \({}^{n}C_r=\frac{n}{r}{}^{n-1}C_{r-1}\)?

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Correct Answer

A. पहले एक marked member चुनें और फिर बाकी (r-1) चुनेंFirst choose one marked member and then choose remaining (r-1)

Explanation

Simple Explanation

Marked member को (n) ways में चुनकर (r) possible marks के overcount को हटाते हैं। परीक्षा में member marking से ऐसी identities समझें। / Choose the marked member in (n) ways and remove overcount of (r) possible marks. In exams understand such identities by member marking.

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Question 356/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

यदि \(^{n}C_{r-1}:{}^{n}C_r=3:5\), तो ratio derivation में कौन-सा expression उपयोगी है?

If \(^{n}C_{r-1}:{}^{n}C_r=3:5\), which expression is useful in the ratio derivation?

Explanation opens after your attempt
Correct Answer

B. \(\frac{{}^{n}C_{r-1}}{{}^{n}C_r}=\frac{r}{n-r+1}\)

Explanation

Simple Explanation

Consecutive combinations में factorial cancel करने पर \(\frac{r}{n-r+1}\) मिलता है। परीक्षा में ratio questions में full values न निकालें। / Canceling factorials in consecutive combinations gives \(\frac{r}{n-r+1}\). In exams do not calculate full values in ratio questions.

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Question 357/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\({}^{n}C_r\) के largest term को पहचानने के लिए कौन-सा relation सबसे उपयोगी है?

Which relation is most useful for identifying the largest term of \({}^{n}C_r\)?

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Correct Answer

A. \(\frac{{}^{n}C_{r+1}}{{}^{n}C_r}=\frac{n-r}{r+1}\)

Explanation

Simple Explanation

Largest term के पास consecutive ratio (1) के आसपास बदलता है। परीक्षा में increasing और decreasing transition देखें। / Near the largest term, the consecutive ratio changes around (1). In exams check the transition from increasing to decreasing.

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Question 358/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) identical balls को (r) distinct boxes में खाली box allowed होने पर बांटने का सूत्र क्या है?

What is the formula for distributing (n) identical balls into (r) distinct boxes when empty boxes are allowed?

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Correct Answer

C. \({}^{n+r-1}C_{r-1}\)

Explanation

Simple Explanation

Stars and bars में (n) stars और (r-1) bars arrange होते हैं। परीक्षा में identical distribution में bars method लगाएं। / In stars and bars, (n) stars and (r-1) bars are arranged. In exams use the bars method for identical distribution.

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Question 359/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

(n) identical balls को (r) distinct boxes में हर box non-empty हो तो formula कैसे बदलता है?

How does the formula change if (n) identical balls are distributed into (r) distinct boxes with every box non-empty?

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Correct Answer

A. \({}^{n-1}C_{r-1}\)

Explanation

Simple Explanation

हर box को पहले (1) ball दें, फिर बाकी (n-r) balls distribute करें। परीक्षा में non-empty condition में पहले minimum allot करें। / Give (1) ball to each box first, then distribute the remaining (n-r) balls. In exams allot the minimum first for non-empty conditions.

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Question 360/600 Hard Mathematics Permutations and Combinations Derivations of formulas and their connections Class 11 Level 65

\({}^{n+r-1}C_r\) और \({}^{n+r-1}C_{n-1}\) का equality किस identity से आती है?

The equality of \({}^{n+r-1}C_r\) and \({}^{n+r-1}C_{n-1}\) comes from which identity?

Explanation opens after your attempt
Correct Answer

B. Complement identity

Explanation

Simple Explanation

दोनों lower indices का sum (n+r-1) है। परीक्षा में stars and bars answers में complementary forms accept करें। / The two lower indices sum to (n+r-1). In exams accept complementary forms in stars and bars answers.

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