If $^{18} C_{r} =^{18} C_{r + 2}$ , Find the value of $^{r} C_{5}$ .

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Solution

${^{18} C}_{r} = {^{18} C}_{r + 2}$
So, $r = r + 2$ or $18 - r = r + 2$
$\times$ $16 = 2 r$
$r = 8$
So, ${^{8} C}_{5} = \frac{8!}{5! 3!} = \frac{6 \times 7 \times 8}{6} = 56.$
Hence, the answer is $56.$

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