The coefficient of $x^{n - 1}$ in the expansion of
$E = {(2 x + 1)}^{n - 1} + {(2 x + 1)}^{n - 2} (x + 1) + . . . . + {(x + 1)}^{n - 1}$ is

2^{n}

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2^{n} - 1

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2^{n} + 1

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2^{2 n}

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Solution

The correct option is C $2^{n} - 1$
Sequence $E$ is in $G . P .$
Therefore, sum of G.P $E = \frac{1}{x} [{(2 x + 1)}^{n} - {(x + 1)}^{n}]$
Now, coefficient of $x^{n - 1}$ is coefficient of $x^{n}$ in ${(2 x + 1)}^{n} - {(x + 1)}^{n}$ .
which is ${^{n} C}_{n} 2^{n} - {^{n} C}_{n} = 2^{n} - 1$

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