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

^{2 n} C_{n - 2}

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^{2 n} C_{n}

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^{2 n} C_{n - 1}

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^{2 n} C_{2 n - 2}

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Solution

The correct option is A $^{2 n} C_{n - 2}$
$(1 + x)^{n} = C_{o} + C_{1} x + C_{2} x^{2} + . . . . . + C_{n} x^{n}$ …………. $(1)$
$(x + 1)^{n} = C_{o} x^{n} + C_{1} x^{n - 1} + . . . . . + C_{n} x^{o}$ …………… $(2)$
Multiplying $(1)$ & $(2)$
$(C_{o} + C_{1} x + C_{2} x^{2} + . . . . + C_{n} x^{n}) \times (C_{o} x^{n} + C_{1} x^{n - 1} + . . . . C_{n} x^{o}) = (1 + x)^{2 n}$
Comparing the coefficients of $x^{n - 2}$ or $x^{n + 2}$ both sides.
We get
$C_{o} C_{2} + C_{1} C_{3} + C_{2} C_{4} + . . . . + C_{n - 1} C_{n}$
$=^{2 n} C_{n - 2}$ or $=^{2 n} C_{n + 1}$ .

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