C_0 C_r + C_1 C_{r+1} + C_2 C_{r+2} + dots + | vedclass

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(A) We know that $(1+x)^n = \sum_{k=0}^{n} C_k x^k$ and $(1+x)^n = \sum_{k=0}^{n} C_k x^{n-k}$.Multiplying these,we consider the coefficient of $x^{n+

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$\frac{(2n)!}{(n-r)!(n+r)!}$

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(A) We know that $(1+x)^n = \sum_{k=0}^{n} C_k x^k$ and $(1+x)^n = \sum_{k=0}^{n} C_k x^{n-k}$.Multiplying these,we consider the coefficient of $x^{n+r}$ in the expansion of $(1+x)^n (1+x)^n = (1+x)^{2n}$.The coefficient of $x^{n+r}$ in $(1+x)^{2n}$ is given by $^{2n}C_{n+r}$.Thus,the sum $C_0 C_r + C_1 C_{r+1} + \dots + C_{n-r} C_n = ^{2n}C_{n+r} = \frac{(2n)!}{(n+r)!(n-r)!}$.

$C_0 C_r + C_1 C_{r+1} + C_2 C_{r+2} + \dots + C_{n-r} C_n =$

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