8,9 Determine the coefficients that will stand by $x^{17}$ and $x^{18}$ after expanding the brackets and combining like terms in the expression
$$
\left(1+x^{5}+x^{7}\right)^{20}
$$
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 permutation of the addends. In one of the 20 factors $1+x^{5}+x^{7}$, we must choose $x^{7}$, and in two of the remaining 19 factors, we must choose $x^{5}$. Therefore, the coefficient of $x^{17}$ is $20 \cdot 19 \cdot 18: 2=3420$.

## Answer

3420 and 0.