Question

If ⁿP₄ = 24. ⁿC₅, then the value of n is ____________.

Answer 1

Given :- ⁿP₄ = 24. ⁿC₅
Since ${}^{n}C_5=\frac{{}^{n}P_5}{5!}\left(\because {}^{n}C_r=\frac{{}^{n}P_r}{r!}\right)$
$$\begin{gathered} \therefore {}^{n}P_4=\frac{24{}^{n}P_5}{5!} \\ \therefore \frac{{}^{n}P_5}{{}^{n}P_4}=\frac{5!}{24}=\frac{5\times 4\times 3\times 2}{24} \\ \mathrm{i}.\mathrm{e}\frac{n!}{\left(n-5\right)!}\times \frac{\left(n-4\right)!}{n!}=5 \end{gathered}$$
i.e n − 4 = 5
i.e n = 9