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Question Number 56042 by Tawa1 last updated on 09/Mar/19

	Answered by math1967 last updated on 09/Mar/19

	${\overset{n}{c}}_{r} = \frac{n!}{r! (n - r)!} \times \frac{(n - r + 1)}{(n - r + 1)} = \frac{n! \times (n - r + 1)}{r \times (r - 1)! (n - r + 1)!}$ $= \frac{n!}{(r - 1)! (n - r + 1)!} \times [\frac{n - r + 1}{r}]$ $= \overset{n}{} c_{r - 1} [\frac{(n - r + 1)}{r}]$
	Commented by Tawa1 last updated on 09/Mar/19

	$God bless you sir$
	Answered by math1967 last updated on 09/Mar/19
	$c_(r ) ^n + ^n c_(r−1) =((n!)/(r!(n−r)!)) +((n!)/((r−1)!(n−r+1)!)) =n!{(1/(r(r−1)!(n−r)!)) +(1/((r−1)!(n−r+1)(n−r)!))} =((n!)/((r−1)!(n−r)!))×{((n−r+1+r)/(r(n−r+1)))} =(((n+1)×n!)/(r(r−1)!(n−r+1)(n−r)!))=(((n+1)!)/(r!(n+1−r)!)) = ^(n+1) c_r$
	${\overset{n}{c}}_{r} + \overset{n}{} c_{r - 1} = \frac{n!}{r! (n - r)!} + \frac{n!}{(r - 1)! (n - r + 1)!}$ $= n! {\frac{1}{r (r - 1)! (n - r)!} + \frac{1}{(r - 1)! (n - r + 1) (n - r)!}}$ $= \frac{n!}{(r - 1)! (n - r)!} \times {\frac{n - r + 1 + r}{r (n - r + 1)}}$ $= \frac{(n + 1) \times n!}{r (r - 1)! (n - r + 1) (n - r)!} = \frac{(n + 1)!}{r! (n + 1 - r)!}$ $= \overset{n + 1}{} c_{r}$
	Commented by Tawa1 last updated on 09/Mar/19

	$God bless you sir$
	Commented by math1967 last updated on 09/Mar/19

	$G O D I S A C I R C L E W H O S E C E N T R E$ $I S E V E R Y W H E R E, B U T W H O S E$ $C I R C U M F E R E N C E I S N O W H E R E$
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