Evaluate

Given : n = 15 and r = 12

To Find: Value of $\frac{n !}{(r !) \times(n-r) !}$ at given $\mathrm{n}$ and $\mathrm{r}$

Formula :

$n !=n \times(n-1) !$

$n !=n \times(n-1) \times(n-2) \ldots \ldots \ldots \ldots 3 \times 2 \times 1$

Let ,

$x=\frac{n !}{(r !) \times(n-r) !}$

Substituting n = 15 and r = 12 in above equation,

$\therefore x=\frac{(15 !)}{(12 !) \times(15-12) !}$

$\therefore x=\frac{(15 !)}{(12 !) \times(3) !}$

By using above formula,

$\therefore x=\frac{15 \times 14 \times 13 \times 12 !}{(12 !) \times(3 \times 2 \times 1)}$

Cancelling (12!) from numerator & denominator,

$\therefore x=\frac{15 \times 14 \times 13}{3 \times 2 \times 1}$

$\therefore \mathrm{X}=455$

Conclusion: Value of $\frac{n !}{(r !) \times(n-r) !}$ at $\mathrm{n}=15$ and $\mathrm{r}=12$ is 6