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If $C_{0}, C_{1}, C_{2}, \dots, C_{n}$ denote the binomial coefficients in the expansion of $(1 + x)^{n},$ then the value of $\sum_{r = 0}^{n} (r + 1) C_{r},$ is

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$n 2^{n}$

$(n + 1) 2^{n - 1}$

$(n + 2) 2^{n - 1}$

$(n + 2) 2^{n - 2}$

answer is C.

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Detailed Solution

$\sum_{r = 0}^{n} {(r + 1)}^{n} C_{r}$

$\begin{array}{l} = \sum_{r = 0}^{n} r \cdot^{n} C_{r} + \sum_{r = 0}^{n}^{n} C_{r} \\ = n \sum_{r = 1}^{n - 1} C_{r - 1} + \sum_{r - 0}^{n}^{n} C_{r} - n \cdot 2^{n - 1} + 2^{n} = (n + 2) 2^{n - 1} \end{array}$

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