Question

prove that (n-1)!/(n-r-1)!+r*(n-1)!/(n-r)!=n!/(n-r)!​

Answer 1

I hope u understand it well

Answer 2

Given :

$\bullet \large \: \tt \frac{(n - 1)!}{n - r - 1)!}+ r. \frac{(n - 1)!}{n - r)!}=\frac{n!}{(n-r)!}$

Solution :

On solving L.H.S.

$\tt \implies \frac{(n - 1)!}{n - r - 1)!}+ r. \frac{(n - 1)!}{n - r)!} \\ \\ \tt \implies(n - 1)! \bigg[ \frac{(n - r}{n - r - 1)! \times (n - r)} + \frac{r}{(n - r)!} \bigg] \\ \\ \implies \tt(n - 1)! \bigg[ \frac{n - r}{(n - r)!} + \frac{r}{(n - r)!} \bigg] \\ \\ \tt \implies(n - 1)! \bigg[ \frac{n - r + r}{(n - r)!} \bigg] \\ \\ \tt \implies \frac{(n - 1)! \times n}{(n - r)!} \\ \\ \tt \implies \frac{n!}{(n - r)!} = R.H.S$

Hence Proved