$C_{0} C_{r} + C_{1} C_{r + 1} C_{2} C_{r + 2} + . . . . . . + C_{n - r} C_{n}$ is equal to

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

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\frac{n!}{r!! (n + r)!}

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\frac{n!}{(n - r)!}

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None of these

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Solution

The correct option is A $\frac{2 n!}{(n - r)! (n + r)!}$
we know that :
$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x + . . .^{n} C_{n} x^{n}$
Also, ${(1 + \frac{1}{x})}^{n} =^{n} C_{0} +^{n} C_{1} (\frac{1}{x}) + . . .^{n} C_{n} {(\frac{1}{x})}^{n}$
The summation asked above
= coefficient of $x^{r}$ in $(1 + x)^{n} (1 + \frac{1}{x})^{n}$
= coefficient of $x^{r}$ in $(1 + x)^{n} {(\frac{x + 1}{x})}^{n}$
= coefficient of $x^{r}$ in ${\frac{(1 + x)}{x^{n}}}^{2 n}$
= coefficient of $x^{n + r}$ in $(1 + x)^{2 n}$
$=_{n + r}^{2 n} C$
$= \frac{2 n!}{(n + r)! (2 n - n - r)!}$
$= \frac{2 n!}{(n + r)! (n - r)!}$

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Similar questions

(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_{o} C_{r} + C_{1} C_{r + 1} + C_{2} C_{r + 2} . . . . . . . . . . . . C_{n - r} C_{n} = 2 n C_{(n + r)}

C_{o}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . . . . . . . . + C_{a}^{2} = {2 n}_{n}^{C}

2 \leq r \leq n, (\frac{n}{r}) + 2 (\frac{n}{r - 1}) + (\frac{n}{r - 2})

m \sum r = 0

^{n + r} C_{n} =

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