Find $^{'} n^{'}$ if $n C_{13} = n C_{12}$ .

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Solution

$n C_{13} = n C_{12}$
$\frac{n!}{13! (n - 13)!} = \frac{n!}{12! (n - 12)!}$
$\frac{1}{13! (n - 13)!} = \frac{1}{12! (n - 12)!}$
$\frac{1}{13 \times 12! (n - 13)!} = \frac{1}{12! (n - 12) \times (n - 13)!}$
$\frac{1}{13} = \frac{1}{n - 12}$
$n - 12 = 13$
$n = 25$

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