10.7. Determine the coefficients that will stand by \(x^{17}\) and \(x^{18}\) after expanding the brackets and combining like terms in the expression

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

The number 17 can be represented as the sum of several 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 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 $20 \cdot \frac{19 \cdot 18}{2}=3420$.