The number of terms in the Multinomial Expansion $(x+\frac{1}{x}+x^2+\frac{1}{x^2})^n$ I am aware that there is a formula to calculate the number of terms in a multinomial expression $(x_1+x_2+x_3+...x_r)^n$, i.e. $^{n+r-1}C_{r-1}$. However, this is in the case when the terms $x_1, x_2, x_3 ... x_r$ are different variables. In my case, the variables are the same; i.e. x, raised to different powers. Can someone please point me in the right direction?

HINT-You have $$(x+\frac{1}{x}+x^2+\frac{1}{x^2})^n=\left(\frac{x^4+x^3+x+1}{x^2}\right)^n=\frac{(x+1)^n(x^3+1)^n}{x^{2n}}$$ Therefore you don't need in this case the multinomial
$$\displaystyle (x_1+x_2+x_3+x_4)^n=\sum_{0 \le k_i\le 4}^{}\binom{n}{k_1,k_2,k_3,k_4}\prod_{i=1}^{4}x_i^{k_i}$$ which has $35$ terms. Anyway you have to calculate the corresponding simplifications in exponents.
