## Task A-4.2.

What is the coefficient of $x^{9}$ in the polynomial $\left(1+x^{3}+x^{6}\right)^{10}$?
## First Solution.

We want to determine the coefficient of $x^{9}$ in the polynomial

$$
\underbrace{\left(1+x^{3}+x^{6}\right)\left(1+x^{3}+x^{6}\right) \cdots\left(1+x^{3}+x^{6}\right)}_{10 \text { times }}
$$

Since using the addends 6 and 3, the number 9 can only be represented as $6+3$ and $3+3+3$, if we multiply all the parentheses in the above expression, $x^{9}$ can be obtained in one of the following two ways:

1) From one of the 10 parentheses, we choose $x^{6}$, from one of the remaining 9 parentheses, we choose $x^{3}$, and from all the other parentheses, we choose 1. This can be done in $10 \cdot 9=90$ ways.
2) From three of the 10 parentheses, we choose $x^{3}$, and from the remaining, we choose 1.

This can be done in $\binom{10}{3}=120$ ways.