The value of the expression $C_{0} C_{2} + C_{1} C_{3} + C_{2} C_{4} + . . . . . + C_{n - 2} C_{n},$ is equal to

^{2 n} C_{n - 1}

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^{2 n} C_{n - 2}

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^{2 n} C_{n}

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None of the above

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Solution

The correct option is A $^{2 n} C_{n - 2}$
$(1 + x)^{n} =^{n} C_{0} +^{n} C_{1} x +^{n} C_{2} x^{2} +^{n} C_{3} x^{3} + . . . . . . . . . . . . . .$ (1)
$(x + 1)^{n} =^{n} C_{0} x^{n} +^{n} C_{1} x^{n - 1} +^{n} C_{2} x^{n - 2} + . . . . . . . . . . . . . .$ (2)
Multiplying Equation(1) and (2), we get
$(1 + x)^{2 n} = ((^{n} C_{0})^{2} + (^{n} C_{1})^{2} + . . . . . . . . . . .) x^{n} + (^{n} C_{0} \cdot^{n} C_{1} +^{n} C_{1} \cdot^{n} C_{2} + . . . . . . . . . . . . .) x^{n - 1} + (^{n} C_{0}^{n} C_{2} +^{n} C_{1}^{n} C_{3} +^{n} C_{2}^{n} C_{4} + . . . . . . . . . . . . . .) x^{n - 2} . . . . . . . . . .$
Comparing co-efficients of $x^{n - 2}$ we get
$^{n} C_{0}^{n} C_{2} +^{n} C_{1}^{n} C_{3} +^{n} C_{2}^{n} C_{4} + . . . . . . . . =^{2 n} C n - 2$

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Similar questions

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

^{2 n} C_{n + 1}

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

^{2 n} C_{n} -^{n} C_{1} .^{2 n - 2} C_{n} +^{n} C_{2} .^{2 n - 4} C_{n} - . . .

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . + C_{n} x^{n}

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

\frac{1}{2}

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