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Factorial

P n = 1+ x n-...

Question

$P (n) = {(1 + x)}^{n} - n x - 1$ , is divisible by

x^{3}

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x^{2}

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any exponent of

x

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none of these

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Solution

The correct option is A $x^{2}$
$P (n) = {(1 + x)}^{n} - n x - 1$
$\Rightarrow P (n) = 1 + n x + n_{C_{2}} x^{2} + . . . + n_{C_{n}} x^{n} - 1 - n x = x^{2} (n_{C_{2}} + n_{C_{3}} x + . . . + n_{C_{n}} x^{n - 2})$
Therefore, $P (n)$ is divisible by $x^{2}$
Ans: B

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