How many ways are there to divide $100$ different balls into $5$ different boxes so the last $2$ boxes contains even number of balls? How many ways are there to divide $100$ different balls into $5$ different boxes so the last $2$ boxes contains even number of balls?
I tried to think about tylor function but got stuck.
Thanks.

Let $g_{e}(x)=\displaystyle\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\right)^3\left(1+\frac{x^2}{2!}+\frac{x^4}{4!}+\cdots\right)^2=(e^x)^3\left(\frac{e^x+e^{-x}}{2}\right)^2$
$\;\;\;\;\;\;\;\;\;\;\;\;=\displaystyle e^{3x}\left(\frac{e^{2x}+2+e^{-2x}}{4}\right)=\frac{1}{4}(e^{5x}+2e^{3x}+e^x)$.
The coefficient of $x^{100}$ is given by $\displaystyle\frac{a_{100}}{100!}=\frac{1}{4}\left(\frac{5^{100}}{100!}+2\cdot\frac{3^{100}}{100!}+\frac{1}{100!}\right)$,  so
$\hspace{1.5 in}$ $\displaystyle a_{100}=\frac{1}{4}(5^{100}+2\cdot3^{100}+1)$.
