Question

If ${}^{n}P_4$ = 360, find the value of n.

Answer 1

ⁿP₄ = 360
$$\begin{gathered} \Rightarrow \frac{n!}{\left(n-4\right)!}=360 \\ \Rightarrow \frac{n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)!}{\left(n-4\right)!}=360 \\ \Rightarrow n\left(n-1\right)\left(n-2\right)\left(n-3\right)=360 \\ \Rightarrow n\left(n-1\right)\left(n-2\right)\left(n-3\right)=6\times 5\times 4\times 3 \\ \mathrm{On}\mathrm{comparing}\mathrm{the}\mathrm{two}\mathrm{sides},\mathrm{we}\mathrm{get}: \\ n=6 \end{gathered}$$