If $^nP4 = 30 imes ^nP2$, find the value of $n$.

Question

If $^nP4 = 30 	imes ^nP2$, find the value of $n$.

Accepted Answer

Using the formula $^nPr = \frac{n!}{(n-r)!}$, we have $\frac{n!}{(n-4)!} = 30 	imes \frac{n!}{(n-2)!}$. 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.

Permutations and Combinations

❓ Sibling Questions in this Chapter