If the coefficient of $x^{4}$ in the expansion of the polynomial $p (x) = (1 + x + x^{2} + x^{3} + . . . . . . . . . + x^{100})^{5}$ is $n$ then the value of $\frac{n}{10}$ is ______.

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Solution

Given polynomial is
$p (x) = (1 + x + x^{2} + x^{3} + . . . . . . . . . + x^{100})^{5}$
$p (x) = {(\frac{1 (1 - x^{100})}{1 - x})}^{5}$
$\Rightarrow p (x) = (1 - x^{100})^{5} (1 - x)^{- 5}$
$\Rightarrow p (x) = (1 - x^{100})^{5} [1 + 5 x + 15 x^{2} + 35 x^{3} + 70 x^{4} + . . .]$
Coefficient of $x^{4}$ in the expansion is 70.
$\frac{n}{10} = \frac{70}{10} = 7$

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