Value of the expression $C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . + (n + 1) C_{n}^{2}$ is

(2 n + 1) ({^{2 n} C}_{n})

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(2 n - 1) ({^{2 n} C}_{n})

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(\frac{n}{2} + 1) ({^{2 n} C}_{n})

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(\frac{n}{2} + 1) ({^{2 n - 1} C}_{n})

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Solution

The correct option is C $(\frac{n}{2} + 1) ({^{2 n} C}_{n})$
Let $S = C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . n C_{n - 1}^{2} + (n + 1) C_{n}^{2} . . . . . . (1)$
using $C_{r} = C_{n - r}$ , we write $(1)$ in the reverse order to obtain
$S = (n + 1) C_{0}^{2} + C_{1}^{2} + . . . .2 C_{n - 1}^{2} + C_{n}^{2} . . . . . . (2)$
Adding $(1)$ and $(2)$ , we get
$2 S = (n + 2) [C_{0}^{2} + C_{1}^{2} + . . . . . . + C_{n}^{2}] = (n + 2) ({^{2 n} C}_{n})$
$\Rightarrow S = (\frac{n}{2} + 1) ({^{2 n} C}_{n})$

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Value of the expression C20+C21+C22+.....+(n+1)C2n is

The correct option is C (n2+1)(2nCn)Let S=C20+C21+C22+.....nC2n−1+(n+1)C2n......(1)using Cr=Cn−r, we write (1) in the reverse order to obtainS=(n+1)C20+C21+....2C2n−1+C2n......(2)Adding (1) and (2), we get2S=(n+2)[C20+C21+......+C2n]=(n+2)(2nCn)⇒S=(n2+1)(2nCn)

Value of the expression $C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + . . . . . + (n + 1) C_{n}^{2}$ is