The value of (C_0 + C_1 )(C_1 + C_2 ).... (C_(n−1) + C_n ) is (1) (((n + 1)^n )/(n!)) ∙ C_1 C_2 .....C_n (2) (((n − 1)^n )/(n!)) ∙ C_1 C_2 .....C_n (3) (((n)^n )/((n + 1)!)) ∙ C_1 C_2 .....C_n (4) (((n)^n )/(n!)) ∙ C_1 C_2 .....C


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Question Number 23856 by Tinkutara last updated on 08/Nov/17

$The value of (C_{0} + C_{1}) (C_{1} + C_{2}) . . . .$ $(C_{n - 1} + C_{n}) is$ $(1) \frac{{(n + 1)}^{n}}{n!} \cdot C_{1} C_{2} . . . . . C_{n}$ $(2) \frac{{(n - 1)}^{n}}{n!} \cdot C_{1} C_{2} . . . . . C_{n}$ $(3) \frac{{(n)}^{n}}{(n + 1)!} \cdot C_{1} C_{2} . . . . . C_{n}$ $(4) \frac{{(n)}^{n}}{n!} \cdot C_{1} C_{2} . . . . . C_{n}$
Commented by prakash jain last updated on 08/Nov/17

$C_{k} + C_{k + 1} = \frac{n!}{k! (n - k)!} + \frac{n!}{(k + 1)! (n - k - 1)!}$ $= \frac{n!}{k! (n - k - 1)!} [\frac{n - k + k + 1}{(k + 1) (n - k)}]$ $= \frac{n + 1!}{(k + 1)! (n - k)!} = \frac{(n + 1) (n!)}{(k + 1) [k! (n - k)!]}$ $= \frac{(n + 1)}{(k + 1)}^{n} C_{k}$
Commented by prakash jain last updated on 08/Nov/17

$\underset{k = 0}{\prod^{n - 1}} (C_{k} + C_{k + 1})$ $= \underset{k = 0}{\prod^{n - 1}} \frac{n + 1}{k + 1} C_{k}$ $= \frac{{(n + 1)}^{n}}{n!} C_{0} C_{1} . . . C_{n - 1}$ $= \frac{{(n + 1)}^{n}}{n!} C_{1} C_{2} . . . C_{n}$
Commented by prakash jain last updated on 08/Nov/17

$Option (1) is correct .$
Commented by Tinkutara last updated on 09/Nov/17

$Thank you very much Sir!$
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