Find the exponential generating function for the number of ways to distribute $r$ distinct objects into five different boxes Find the exponential generating function for the number of ways to distribute $r$ distinct objects into five different boxes when $b_1<b_2\le 4$, where $b_1,b_2$ are the numbers of objects in boxes $1$ and $2$, respectively.
I understand that boxes $3, 4$, and $5$ will all just have the fundamental exponential generating function and that their combined generating function will be $e^{3x}$. What I don't understand is the first two boxes with the equality parameter given. 
Can someone explain how to do this?


Three boxes have no restrictions with respect to the objects which results in $e^{3x}$. The restriction $b_1<b_2\leq 4$ of the other objects together with the objects of the three boxes is encoded as
  \begin{align*}
e^{3x}&\left(\left(x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}\right)+x\left(\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}\right)\right.\\
&\quad+\left.\frac{x^2}{2!}\left(\frac{x^3}{3!}+\frac{x^4}{4!}\right)
+\frac{x^3}{3!}\left(\frac{x^4}{4!}\right)\right)\\
&=e^{3x}\left(\frac{x}{1!}+\frac{x^2}{2!}+\frac{4x^3}{3!}+\frac{5x^4}{4!}+
\frac{15x^5}{5!}+\frac{15x^6}{6!}+\frac{35x^7}{7!}\right)
\end{align*}

