Question

If ${}^n{P}_5:n-1P_5=3:2$ then find $n$

Answer 1

$\left(\frac{{}^n{P}_5}{{}^{n-1}{P}_5}\right)=\left(\frac{3}{2}\right)\left(\frac{\left(\frac{n!}{(n-5)!}\right)}{\left(\frac{(n-1)!}{(n-1-5)!}\right)}\right)=\left(\frac{3}{2}\right)\left(\frac{n!(n-6)!}{(n-1)!(n-5)!}\right)=\left(\frac{3}{2}\right)\left(\frac{n(n-1)!\cdot (n-6)!}{(n-1)!\cdot (n-5)(n-6)!}\right)=\left(\frac{3}{2}\right)2x=3(x-5)2x=3x-15\therefore n=15$