If P (n, 5) = 20. P(n, 3), find n ?

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#### Solution

P (n, 5) = 20. P(n, 3)

\[\Rightarrow \frac{n!}{(n - 5)!} = 20 \times \frac{n!}{(n - 3)!}\]

\[ \Rightarrow \frac{n!}{n!} = 20 \times \frac{(n - 5)!}{(n - 3)!}\]

\[ \Rightarrow 1 = 20 \times \frac{(n - 5)!}{(n - 3)(n - 4)(n - 5)!}\]

\[ \Rightarrow (n - 3)(n - 4) = 20\]

\[ \Rightarrow (n - 3)(n - 4) = 5 \times 4\]

\[\text{On comparing the two sides, we get}: \]

\[ \Rightarrow n - 3 = 5\]

\[ \Rightarrow n = 8\]

Concept: Factorial N (N!) Permutations and Combinations

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