Prove that: $C (n, r) + C (n, r - 1) = C (n + 1, r)$

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Solution

$C (n, r) + c (n, r - 1) =^{n} C_{r} +^{n} C_{r - 1}$
$= \frac{n!}{r! n - r!} + \frac{n!}{r - 1! n - r + 1!}$
$= \frac{n! (n - r + 1) + (r)!}{r! n - r + 1!}$
$= \frac{n + 1!}{r! (n + 1) - r!} =^{n + 1} C_{r}$
$= c (n + 1, r)$
Hence proved.

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