Question

Find $n$.
$\frac{n!}{3!(n-5)!}:\frac{n!}{5!(n-7)!}=10:3$

Answer 1

We have,

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

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

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

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

$\Rightarrow \frac{5\times 4\times 3!(n-7)(n-6)(n-5)!}{3!(n-5)!}=\frac{10}{3}$

$\Rightarrow \frac{20(n-7)(n-6)}{1}=\frac{10}{3}$

$\Rightarrow 3(n-7)(n-6)=\frac{10}{20}$

$\Rightarrow 6(n^2-7n-6n+42)=1$

$\Rightarrow 6n^2-78n+252=1$

$\Rightarrow 6n^2-78n+251=0$

Comparing that,

$a{n}^2+bn+c=0$

And we get,

$a=6,b=-78,c=251$

Then, using quadratic formula

$n=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$n=\frac{78\pm \sqrt{{(-78)}^2-4\times 6\times 251}}{2\times 6}$

$n=\frac{78\pm \sqrt{6084-6024}}{12}$

$n=\frac{78\pm \sqrt{60}}{12}$

$n=\frac{78\pm 2\sqrt{15}}{12}$

$n=\frac{39\pm \sqrt{15}}{6}$

Hence, this is the answer.