Value of the expression
$C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . + (C_{0} + C_{1} + . . . . + C_{n - 1})$ is

2^{n - 1}

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n 2^{n - 2}

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n 2^{n - 1}

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2 n

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Solution

The correct option is B $n 2^{n - 1}$
Simplifying, we get
$n^{n} C_{0} + (n - 1)^{n} C_{1} + (n - 2)^{n} C_{2} + . . . (n - n)^{n} C_{n}$
$= n [^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + . . .^{n} C_{n}] - [^{n} C_{1} + 2^{n} C_{2} + 3^{n} C_{3} + . . . n^{n} C_{n}]$
$= n 2^{n} - \frac{d (1 + x)^{n}}{d x} |_{x = 1}$
$= n 2^{n} - n 2^{n - 1}$
$= n 2^{n - 1}$

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