3-ча 1. Determine the coefficients that will stand by $x^{17}$ and $x^{18}$ after expanding the brackets and combining like terms in the expression

$$
\left(1+x^{5}+x^{7}\right)^{20}
$$
Solution 1. The number 18 cannot be represented as the sum of the numbers 5 and 7, so the coefficient of $x^{18}$ will be zero.

The number 17 can be represented as the sum of the numbers 5 and 7 as follows: $17=7+5+5$; this representation is unique up to the order of the terms. In one of the 20 expressions $1+x^{5}+x^{7}$, we must choose $x^{7}$, and in two of the 19 remaining such expressions, we must choose $x^{5}$. Therefore, the coefficient of $x^{17}$ is $20 \cdot \frac{19 \cdot 18}{2}=3420$.