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Question Number 85228 by mathocean1 last updated on 20/Mar/20

Commented by mathocean1 last updated on 20/Mar/20

$p l e a s e h e l p m e . . .$
Commented by abdomathmax last updated on 20/Mar/20
$C_n ^p −C_(n−1) ^p =((n!)/(p!(n−p)!))−(((n−1)!)/(p!(n−p−1)!)) =((n(n−1)!)/(p(p−1)!(n−p)!))−(((n−1)!(n−p))/(p(p−1)!(n−p)!)) =(((n−1)!)/((p−1)!(n−p)!)){(n/p)−((n−p)/p)} =(((n−1)!)/((p−1)!(n−p)!))×1 =C_(n−1) ^(p−1) ⇒ C_n ^p =C_(n−1) ^p +C_(n−1) ^(p−1)$
$C_{n}^{p} - C_{n - 1}^{p} = \frac{n!}{p! (n - p)!} - \frac{(n - 1)!}{p! (n - p - 1)!}$ $= \frac{n (n - 1)!}{p (p - 1)! (n - p)!} - \frac{(n - 1)! (n - p)}{p (p - 1)! (n - p)!}$ $= \frac{(n - 1)!}{(p - 1)! (n - p)!} {\frac{n}{p} - \frac{n - p}{p}}$ $= \frac{(n - 1)!}{(p - 1)! (n - p)!} \times 1$ $= C_{n - 1}^{p - 1} \Rightarrow C_{n}^{p} = C_{n - 1}^{p} + C_{n - 1}^{p - 1}$
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