`_20 C_15` Find the binomial coefficient.

Wednesday, October 12, 2016

Use the formula $\displaystyle_{n}{C}_{{r}}=\frac{{{n}!}}{{{r}!{\left({n}-{r}\right)}!}}$

$\displaystyle_{20}{C}_{{15}}=\frac{{{20}!}}{{{15}!{\left({20}-{15}\right)}!}}$

$\displaystyle=\frac{{{20}\cdot{19}\cdot{18}\cdot{17}\cdot{16}\cdot{15}!}}{{{15}!{5}!}}$

$\displaystyle=\frac{{{20}\cdot{19}\cdot{18}\cdot{17}\cdot{16}}}{{{5}\cdot{4}\cdot{3}\cdot{2}\cdot{1}}}$

$\displaystyle={15504}$

Use the formula $\displaystyle_{n}{C}_{{r}}=\frac{{{n}!}}{{{r}!{\left({n}-{r}\right)}!}}$

$\displaystyle_{20}{C}_{{15}}=\frac{{{20}!}}{{{15}!{\left({20}-{15}\right)}!}}$

$\displaystyle=\frac{{{20}\cdot{19}\cdot{18}\cdot{17}\cdot{16}\cdot{15}!}}{{{15}!{5}!}}$

$\displaystyle=\frac{{{20}\cdot{19}\cdot{18}\cdot{17}\cdot{16}}}{{{5}\cdot{4}\cdot{3}\cdot{2}\cdot{1}}}$

$\displaystyle={15504}$

Notes