Question

If ^(n−1)P3 :

^n P4 = 1 : 10, find n.

Answer 1

Answer:

Given,

$\: \: \: \: ^ {(n - 1) }P_3: \: ^nP_4 = 1:10 \\ \sf \implies \: \frac{^ {(n - 1) }P_3}{^nP_4} = \frac{1}{10} \\ \sf \implies \: \frac{ \frac{(n - 1)!}{(n - 1 - 3)! \: } }{ \frac{n!}{(n - 4)!} } = \frac{1}{10} \\ \sf \implies \: \frac{(n - 1)!}{ \cancel{(n - 4)!}} \times \frac{ \cancel{(n - 4)!}}{n!} = \frac{1}{10} \\ \sf \implies \: \frac{(n - 1)!}{n!} = \frac{1}{10} \\ \sf \implies \: \frac{ \cancel{(n - 1)!}}{n \cancel{(n - 1)!}} = \frac{1}{10} \\ \sf \implies \: \frac{1}{n} = \frac{1}{10} \\ \therefore \: \: \sf \: n = 10$

Note :

$> \: \: \: ^nP_r = \frac{n!}{(n - r)!} \: \\ > \: n! = n(n - 1)(n - 2)...(n - r + 1).(n - r)!$