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Q.

The value of $\sum_{r = 0}^{n} \sum_{s = 1}^{n}^{n} C_{s}^{s} C_{r}$ is

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a

$3^{n} - 1$

b

$3^{n} + 1$

c

$3^{n}$

d

none of these

answer is A.

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Detailed Solution

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$\begin{array}{l} \sum_{r = 0}^{n} \sum_{s = 1}^{n}^{n} {C_{s}}^{s} C_{r} \Rightarrow \sum_{r = 0}^{n}^{n} C_{1} (^{1} C_{r}) +^{n} {C_{2}}^{2} C_{r} + \dots \dots^{n} {C_{n}}^{n} C_{r} \\ {\Rightarrow=}^{n} C_{1} (2) +^{n} C_{2} 2^{2} +^{n} C_{3} 2^{3} + \dots .^{n} C_{n} 2^{n} \\ \Rightarrow= (1 + 2)^{n} - 1 = 3^{n} - 1 \end{array}$

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The value of ∑r=0n ∑s=1n nCssCr is