Solve the following
Question:

If nP4 : nP5 = 1 : 2, then n =__________.

Solution:

${ }^{n} P_{4}:{ }^{n} P_{5}=1: 2$

i. e $\frac{{ }^{n} P_{4}}{{ }^{n} P_{5}}=\frac{1}{2}$

$\frac{n !}{(n-4) !} \times \frac{(n-5) !}{n !}=\frac{1}{2}$

$\frac{n !(n-5) !}{(n-4)(n-5) ! n !}=\frac{1}{2}$

$2=n-4$

i. e $n=6$