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Combination

Prove that ...

Question

Prove that ${^{n} C}_{n - r} - {^{n} C}_{r} = 0$ .

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Solution

Consider $^{n} C_{n - r} -^{n} C_{r}$
$= \frac{n!}{(n - (n - r))! (n - r)!} - \frac{n!}{(n - r)! r!}$

$= \frac{n!}{r! (n - r)!} - \frac{n!}{(n - r)! r!}$
$= 0$

Hence, $^{n} C_{n - r} -^{n} C_{r} = 0$ .

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