If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . + C_{n} x^{n}$ , then $C_{0} C_{2} + C_{1} C_{3} + C_{2} C_{4} + . . . + C_{n - 2} C_{n} =$

\frac{(2 n)!}{(n!)^{2}}

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\frac{(2 n)!}{(n - 1)! (n + 1)!}

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

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none of these

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Solution

The correct option is C $\frac{(2 n)!}{(n - 2)! (n + 2)!}$
Consider the following series.
$^{m} C_{0}^{n} C_{r} +^{m} C_{1}^{n} C_{r - 1} + . . .^{m} C_{r}^{n} C_{0}$
=Coefficient of $x^{r}$ in $(1 + x)^{m} . (x + 1)^{n}$
=coefficient of $x^{r}$ in $(1 + x)^{m + n}$
$=^{m + n} C_{r}$
In the above case, $r = n - 2$ and $m = n$
Hence
$^{2 n} C_{n - 2}$
$= \frac{2 n!}{(2 n - (n - 2))! (n - 2)!}$
$= \frac{2 n!}{(n + 2)! (n - 2)!}$

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Similar questions

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3} + \dots + C_{n} x^{n},

C_{0} C_{2} + C_{1} C_{3} + C_{2} C_{4} + \dots + C_{n - 2} C_{n} =

C_{0} C_{2} + C_{1} C_{3} + C_{2} C_{4} + C_{n - 2} C_{n} = \frac{(2 n)!}{(n - 1)! (n = 1)!}

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If $(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . + C_{n} x^{2}$ , then

$C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + C_{3}^{2} + . . . . . . . . . . + C_{n}^{2}$ =

I f (1 + x)^{n} = C_{0} + c_{1} x + . . . + C_{n} x^{n} +, t h e n \frac{C_{1}}{C_{0}} + \frac{2 C_{2}}{C_{1}} + \frac{3 C_{3}}{C_{2}} + . . . . . + \frac{n C_{n}}{C_{n - 1}} i s

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