MCQMathematics

MP Board · Class 11 · Mathematics · Permutations and CombinationsIf $^nP4 = 30 \times ^nP2$, find the value of $n$.

Q: If $^nP4 = 30 \times ^nP2$, find the value of $n$.

Using the formula $^nPr = \frac{n!}{(n-r)!}$, we have $\frac{n!}{(n-4)!} = 30 \times \frac{n!}{(n-2)!}$. Simplifying this gives $\frac{1}{(n-4)!} = \frac{30}{(n-2)(n-3)(n-4)!}$, which implies $(n-2)(n-3) = 30$. Solving $n^2 - 5n + 6 = 30 \implies n^2 - 5n - 24 = 0 \implies (n-8)(n+3) = 0$. Since $n$ must be positive, $n = 8$. Thus, option D is correct.

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Total Attempts: 8

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Step-by-Step Solution

Using the formula $^nP_r = \frac{n!}{(n-r)!}$, we have $\frac{n!}{(n-4)!} = 30 \times \frac{n!}{(n-2)!}$. Simplifying this gives $\frac{1}{(n-4)!} = \frac{30}{(n-2)(n-3)(n-4)!}$, which implies $(n-2)(n-3) = 30$. Solving $n^2 - 5n + 6 = 30 \implies n^2 - 5n - 24 = 0 \implies (n-8)(n+3) = 0$. Since $n$ must be positive, $n = 8$. Thus, option D is correct.

Detailed Options Breakdown

Option : 6

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Permutations and Combinations.

Option 1: 5

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Permutations and Combinations.

Option 2: 7

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Permutations and Combinations.

Option 3: 8 (Correct Answer)

Correct choice. Refer to the step-by-step verified solution guidelines above for details.

💡 Study Guide: This question tests core syllabus concepts from Permutations and Combinations. For formulas, key summaries, and mock exam reference guides, read the full Permutations and Combinations Revision Notes.

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