Calculate the coefficient of $x^{100}$ in the polynomial $\left(1+x+x^{2}+\ldots+x^{100}\right)^{3}$ after combining all like terms.

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The coefficient of $x^{100}$ is equal to the number of solutions of the equation $p+q+r=100$ in non-negative integers.

## Solution

Multiplying the polynomial $1+x+x^{2}+\ldots+x^{100}$ by itself twice, we get the sum of monomials of the form $x^{p} x^{q} x^{r}$, where $p$, $q, r$ independently take values from 0 to 100. Therefore, the coefficient of $x^{100}$ is equal to the number of solutions of the equation $p+q+r=100$ in non-negative integers. This number, in turn, is equal to the number of ways to distribute 100 balls into three boxes. See problems $\underline{30717}$ b) and $\underline{60406}$ b).

## Answer

$C_{102}^{2}=5151$