Question

If 5 P(4, n) = 6. P (5, n − 1), find n.

Answer 1

5 P(4, n) = 6. P (5, n − 1)
5 ⁴P_n = 6⁵P_n$-$1
$$\begin{gathered} \Rightarrow 5\times \frac{4!}{\left(4-n\right)!}=6\times \frac{5!}{\left(5-n+1\right)!} \\ \Rightarrow 5\times \frac{\left(6-n\right)!}{\left(4-n\right)!}=6\times \frac{5!}{4!} \\ \Rightarrow 5\times \frac{\left(6-n\right)\left(6-n-1\right)\left(6-n-2\right)!}{\left(4-n\right)}=6\times \frac{5\times 4!}{4!} \\ \Rightarrow 5\times \frac{\left(6-n\right)\left(5-n\right)\left(4-n\right)!}{\left(4-n\right)}=6\times 5 \\ \Rightarrow \left(6-n\right)\left(5-n\right)=6 \\ \Rightarrow \left(6-n\right)\left(5-n\right)=3\times 2 \\ \mathrm{On}\mathrm{comparing}\mathrm{the}\mathrm{LHS}\hspace{0.17em}\mathrm{and}\text{the}\mathrm{RHS},\mathrm{we}\mathrm{get}: \\ \Rightarrow 6-n=3 \\ \Rightarrow n=3 \end{gathered}$$