With usual notations ;
$C_{0} C_{r} + C_{1} C_{r + 1} . . . . + C_{n - r} C_{n} =$

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Solution

$\begin{matrix} w e k n o w t h a t {(1 + x)}^{n} =^{n} C_{0} +^{n} C_{1} x + . . . . +^{n} C_{n} x^{n} {(1 + \frac{1}{x})}^{n} =^{n} C_{0} +^{n} C_{1} (\frac{1}{x}) + . . . . +^{n} C_{n} {(\frac{1}{x})}^{n} g i v e n s u m m a t i o n = c o e f f i c i e n t o f x^{r} i n {(1 + x)}^{n} {(1 + \frac{1}{x})}^{n} = c o e f f i c i e n t o f x^{r} i n \frac{{(1 + x)}^{2 n}}{x^{n}} = c o e f f i c i e n t o f x^{n + r} i n {(1 + x)}^{2 n} =^{2 n} C_{n + r} = \frac{2 n!}{(n + r)! (2 n - n - r)!} = \frac{2 n!}{(n + r)! (n - r)!} w h i c h i s t h e r e q u i r e d a n s w e r . \end{matrix}$

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Similar questions

C_{0} C_{r} + C_{1} C_{r + 1} C_{2} C_{r + 2} + . . . . . . + C_{n - r} C_{n}

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . + C_{n} x^{n}

C_{0} C_{r} + C_{1} C_{r + 1} + C_{2} C_{r + 2} + . . . C_{n - r} C_{n} =

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . + C_{n} x^{n}

C_{0} C_{r} + C_{1} C_{r + 1} + . . . . . . + C_{n - r} C_{n} = \frac{(2 n)!}{(n + r)! (n - r)!}

C_{0}, C_{1}, C_{2}, \dots, C_{n}

{(1 + x)}^{n}

C_{0} C_{r} + C_{1} C_{r - 1} + C_{2} C_{r - 2} + \dots + C_{n - r} C_{r} +

C_{0} C_{r} + C_{1} C_{r + 1} + C_{2} C_{r + 2} + C_{r} C_{0} = \frac{(2 n)!}{(n - 1)! (n = 1)!}

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