The value of ∑r=0n ∑s=1n nCssCr is

∑r=0n ∑s=1n nCssCr⇒∑r=0n nC1 1Cr+nC22Cr+……nCnnCr⇒=nC1(2)+nC222+nC323+….nCn2n⇒=(1+2)n−1=3n−1

The value of ∑r=0n ∑s=1n nCssCr is

∑r=0n ∑s=1n nCssCr⇒∑r=0n nC1 1Cr+nC22Cr+……nCnnCr⇒=nC1(2)+nC222+nC323+….nCn2n⇒=(1+2)n−1=3n−1

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

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$3^{n} - 1$

$3^{n}$

none of these

$3^{n} + 1$

answer is A.

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Detailed Solution

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