If C (n, 12) = C (n, 8), then C (22, n) is equal to
(a) 231
(b) 210
(c) 252
(d) 303

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Solution

(a) 231
$^{n} C_{12} =^{n} C_{8}$
$\Rightarrow n = 12 + 8 = 20$ [∵ $^{n} C_{x} =^{n} C_{y} \Rightarrow n = x + y or x = y$ ]

Now, $^{22} C_{n} =^{22} C_{20}$

$= \frac{22}{2} \times \frac{21}{1}$
$= 231$

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