5.9 Expand the expression $\left(x^{7}+x^{5}+1\right)^{20}$ and combine like terms, to get a polynomial. Find the coefficients of $x^{17}$ and $x^{18}$ in this polynomial.
[Solution] Since $17=5 \times 2+7 \times 1$, the coefficient of $x^{17}$ is
$$
\frac{20!}{2!(20-2-1)!}=\frac{20!}{2!\cdot 17!}=\frac{20 \cdot 19 \cdot 18}{2}=3420 \text {. }
$$

Since 18 cannot be expressed as the sum of several 5s and 7s, this polynomial does not contain a term of $x^{18}$, or in other words, the coefficient of $x^{18}$ is 0.