<s>[INST] 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?
 [/INST] 
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*}

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