$C_{0} . C_{2} + C_{1} . C_{3} + C_{2} . C_{4} + . . . . . . . . + C_{n - 2} . C_{n}$

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

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\frac{(2 n)!}{(n - 3)! (n + 3)!}

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

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Solution

The correct option is D $\frac{(2 n)!}{(n - 2)! (n + 2)!}$
We have
$C_{o} + C_{1} x + C_{2} x^{2} + . . . . . + C_{n} x^{n} = (1 + x)^{n}$ …….. $(1)$
Also
$C_{o} x^{n} + C_{2} x^{n - 2} + . . . . . + C_{n} = (x + 1)^{n}$ …….. $(2)$
Multiplying $(1)$ & $(2)$
$(C_{o} + C_{1} x + C_{2} x^{2} + . . . . . + C_{n} x^{n}) \times (C_{o} x^{n} + C_{2} x^{n - 2} + . . . . + C_{n}) = (1 + x)^{2 n}$ ……….. $(3)$
Equating coefficients of $x^{n - r}$ from both sides of $(3)$ , we get
$C_{o} C_{r} + C_{1} C_{r + 1} + C_{2} C_{r + 2} + . . . . . + C_{n - r} C_{n}$
$=^{2 n} C_{n - r} = \frac{(2 n)!}{(n - r)! (n + 2)!}$
$^{2 n} C_{n - 2} = \frac{(2 n)!}{(n - 2)! (n + 2)!}$
Option C is correct.

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