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In the expansion of the following expression $1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$ the coefficient of $x^{k} (0 \leq k \leq n)$ is

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n+1Ck+1

nCk

nCn−k−1

None of these

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Correct option is A

The given expression is1+(1+x)+(1+x)2+…+(1+x)n being in GP. LetS=1+(1+x)+(1+x)2+…+(1+x)n=(1+x)n+1−1(1+x)−1=x−1(1+x)n+1−1∴ The coefficient of xk in S. = The coefficient of xk+1 in (1+x)n+1−1=n+Ck+1

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In the expansion of the following expression 1+(1+x)+(1+x)2+…+(1+x)n the coefficient of xk (0≤k≤n) is

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In the expansion of the following expression $1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}$ the coefficient of $x^{k} (0 \leq k \leq n)$ is