Question

Solve : ${}^{15}{P}_{n-1}:{}^{16}{P}_{n-1}=3:4$

Answer 1

$\frac{{}^{15}{P}_{n-1}}{{}^{16}{P}_{n-1}}=\frac{3}{4}$

$\Rightarrow \frac{\frac{15!}{[15-(n-1\left)\right]!}}{\frac{16!}{(16-(n-1\left)\right)!}}=\frac{3}{4}$

$\Rightarrow \frac{15!}{16\times 15!}\times \frac{(16+1-n)!}{(15+1-n)!}=\frac{3}{4}$

$\Rightarrow \frac{1}{16}\times \frac{(17-n)!}{(16-n)!}=\frac{3}{4}$

$\Rightarrow \frac{(17-n)!}{(16-n)!}=12$

$\Rightarrow \frac{(17-n)!}{(16-n)!}=\frac{12!}{11!}$

$\Rightarrow 17-n=12$

$\Rightarrow n=5$

$\therefore n=5$ if $\frac{{}^{15}{P}_{n-1}}{{}^{16}{P}_{n-1}}=\frac{3}{4}$.