If ${(1 + x)}^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . . . + C_{n} x^{R}$ , then the sum
$C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . + (C_{0} + C_{1} + C_{2} + . . . . . + C_{n - 1}$ is

n . 2^{n + 1}

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n . 2^{n - 1}

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(n - 1) . 2^{n - 1}

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(n + 1) . 2^{n + 1}

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Solution

The correct option is B $n . 2^{n - 1}$
Simplifying, we get
$n^{n} C_{0} + (n - 1)^{n} C_{1} + (n - 2)^{n} C_{2} + . . . (n - (n - 1))^{n} C_{n - 1} + (n - n)^{n} C_{n}$
$= n [^{n} C_{0} +^{n} C_{1} + . . .^{n} C_{n}] - [1.^{n} C_{1} + 2^{n} C_{2} + . . . n^{n} C_{n}]$
$= n {.2}^{n} - n 2^{n - 1}$
$= n 2^{n - 1} (2 - 1)$
$= n 2^{n - 1}$ .

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