If ${^{18} C}_{r} = {^{18} C}_{r + 2}$ , find ${^{r} C}_{5}$ .

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Solution

$^{18} C_{r} =^{18} C_{r + 2}$

$\Rightarrow \frac{18!}{(18 - r)! r!} = \frac{18!}{(16 - r)! (r + 2)!}$

By solving above equation we get, $r = 8$
Using this $r$ ,
$\Rightarrow^{8} C_{5} = 56$

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