The value of $^{2} P_{1} +^{3} P_{1} + . . . . . +^{n} P_{1}$ is equal to

\frac{n^{2} - n + 2}{2}

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\frac{n^{2} + n + 2}{2}

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\frac{n^{2} + n - 1}{2}

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\frac{n^{2} - n - 1}{2}

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\frac{n^{2} + n - 2}{2}

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Solution

The correct option is D $\frac{n^{2} + n - 2}{2}$
The value of $^{2} P_{1} +^{3} P_{1} + \dots \dots +^{n} P_{1}$ is
$= \frac{2!}{1!} + \frac{3!}{2!} + \dots \dots + \frac{n!}{(n - 1)!}$
$= 2 + 3 + 4 + \dots \dots + n$
$= \frac{n (n + 1)}{2} - 1$
$= \frac{n^{2} + n - 2}{2}$

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