Question

Solve for $n$ $\frac{n!}{3!\left(n-5\right)!}:\frac{n!}{5!\left(n-7\right)!}=10:3$

Answer 1

Given that:

$\frac{n!}{3!(n-5)!}\times \frac{5!(n-7)!}{n!}=\frac{10}{3}$

On solving the expression

$\frac{20}{(n-5)(n-6)}=\frac{10}{3}$

$60=10(n^2-6n-5n+30)$

$n^2-11n+30=6$

$n^2-11n+24=0$

$n^2-8n-3n+24=0$

$n(n-8)-3(n-8)=0$

$(n-8)(n-3)=0$

$n=3,8$

Hence, this is the answer.