Multinomial Expansion-Example

Question

Is coefficient of $x^{20}$ in $(1-x+x^2)^{20}$ and $(1+x-x^2)^{20}$ same? Can someone tell me how should i apply multinomial theorem to this problem?

Answer 1

The terms in $x^{20}$ in $(1 + a x + b x^2)^{20}$ are ${20 \choose j,20-2j,j} a^{20-2j} b^j x^{20}$ where $j$ goes from $0$ to $10$. If you replace $a$ by $-a$ and $b$ by $-b$, you change the sign of the terms where $j$ is odd, while leaving those with even $j$ the same. Now each of these terms has the same sign as $b$. Since there are some such terms, and they are nonzero and have the same sign, the coefficients must be different.

More generally, if $a , b$ are real and nonzero and $n$ is a positive integer, the coefficient of $x^n$ in $(1+a+bx^2)^n$ is always different from the coefficient of $x^n$ in $(1-a-bx^2)^n$.