Prove that: n! (n-r)! = n(n-1)(n-2).....(n(n-r)) - Vidyadip

Question

Prove that:$\frac{\text{n!}}{(\text{n-r})!}= \text{n}(\text{n}-1)(\text{n}-2).....(\text{n}(\text{n}-\text{r}))$

✓

Answer

We have, $\text{L.H.S.}=\frac{\text{n!}}{(\text{n-r})!}$ $=\frac{\text{n}(\text{n}-1)(\text{n}-2)(\text{n}-3)...(\text{n}-\text{r}+2)(\text{n}-\text{r}+1)(\text{n}-\text{r})!}{(\text{n}-\text{r})!}$ = n(n - 1)(n - 2)(n - 3).....(n - r + 2)(n - r + 1 )) = n(n - 1)(n - 2)(n - 3).....((n - (r - 2))(n - (r - 1 )) = n(n - 1)(n - 2)(n - 3).....(n - (r - 1))= R.H.S.
$\therefore$ L.H.S. = R.H.S.Hence proved.

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