Question: I am attempting to find the coefficient of $x^5$ in the expression $(2 + x - x^2)^5$ by using the multinomial theorem. I was capable of performing this task only when the number of elements is 2 (using the binomial theorem), however, I am stuck on this one.

Answer: 
Hint: To get $x^5$, the $x^2$ can only be raised to the power $0, 1$ or $2$, so using the multinomial theorem, you only want to look at the multinomial coefficients
$${{5}\choose{0,5,0}}, {{5}\choose{1,3,1}}, {{5}\choose{2,1,2}}.$$
Note that $x^5=(1)^2(x)^1(x^2)^2=(1)^1(x)^3(x^2)^1=(1)^0(x)^5(x^2)^0$.

The coefficient is $\displaystyle \binom{5}{2}(2)^2\binom{3}{1}(1)^1(-1)^2+\binom{5}{1}(2)^1\binom{4}{3}(1)^3(-1)^1+\binom{5}{0}(2)^0\binom{5}{5}(1)^5(-1)^0=81$