293. In which of the expressions

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

will the coefficient of \(x^{20}\) be greater after expanding the brackets and combining like terms?
293. By expanding the brackets and combining like terms in the two expressions under consideration, we will obtain two polynomials in terms of $x$. Now let's replace $x$ with $-x$ in these expressions. This means we will also have to replace $x$ with $-x$ in the resulting polynomials, i.e., keep the same coefficients for $x$ in even powers and change the signs of the coefficients for $x$ in odd powers. In particular, the coefficients of $x^{20}$ do not change during this operation. Thus, we see that our two polynomials have the same coefficients for $x^{20}$ as the polynomials obtained by expanding the brackets and combining like terms in the expressions $\left(1+x^{2}+x^{3}\right)^{1000}$ and $\left(1-x^{2}-x^{3}\right)^{1000}$.

It is clear that the first of these new polynomials has a larger coefficient for $x^{20}$. Indeed, when expanding the brackets in the first expression, we will obtain only positive coefficients for various powers of $x$, and when combining like terms, all these coefficients will add up. In the second expression, when expanding the brackets, we will obtain coefficients for various powers of $x$ that have the same absolute values as the coefficients of the first polynomial, but their signs may differ, and when combining like terms, the coefficients will decrease.

Thus, after expanding the brackets and combining like terms in the expressions $\left(1+x^{2}-x^{3}\right)^{1000}$ and $\left(1-x^{2}+x^{3}\right)^{1000}$, we will obtain a larger coefficient for $x^{20}$ in the first one than in the second.