10.32. In which of the expressions:

$$
\left(1-x^{2}+x^{3}\right)^{1000}, \quad\left(1+x^{2}-x^{3}\right)^{1000}
$$

after expanding the brackets and combining like terms, is the coefficient of \(x^{20}\) greater?
10.32. Answer: in the expression $\left(1+x^{2}-x^{3}\right)^{1000}$. Let $P(x)=$ $=\left(1-x^{2}+x^{3}\right)^{1000}$ and $Q(x)=\left(1+x^{2}-x^{3}\right)^{1000}$. The coefficient of $x^{20}$ in the polynomial $P(x)$ is the same as in $P(-x)=\left(1-x^{2}-x^{3}\right)^{1000}$, and the coefficient of $x^{20}$ in the polynomial $Q(x)$ is the same as in $Q(-x)=\left(1+x^{2}+x^{3}\right)^{1000}$. It is clear that the coefficient of $x^{20}$ in the polynomial $\left(1+x^{2}+x^{3}\right)^{1000}$ is greater than in the polynomial $\left(1-x^{2}-x^{3}\right)^{1000}$. Indeed, in the first polynomial, the term $p_{20} x^{20}$ is the sum of several terms of the form $\left(x^{2}\right)^{n}\left(x^{3}\right)^{m}$, where $2 n+3 m=20$, while in the second polynomial, the term $q_{20} x^{20}$ is the sum of the same terms but with the sign $(-1)^{n+m}$. In the second case, there are terms with a negative sign, for example, when $m=2$ and $n=7$.