A teacher has $n$ students and breaks the students up into some number of groups. Within each group, they assign one student to be a president and another to be a vice president. Let $t_n$ be the number of ways the teacher can proceed. What's the exponential generating function of $t_n$?
Is this similar to a subline problem where you break the lines in 2 lanes? I'm not sure how to do this one
Where you have a set-up where you partition your set into subsets and impose a given structure on each of them, the EGF of the total number of arrangements satisfies $A(x)=\exp(B(x))$ where $B(x)$ is the EGF of the number of structures that can be put on the subsets.
Here we take an $k$-element set, and designate two of its elements. The number of ways of doing so is $b_k=k(k-1)$. So $$B(x)=\sum_{k=0}^\infty b_k\frac{x^k}{k!}=\sum_{k=0}^\infty\frac{ k(k-1)x^k}{k!}=\sum_{k=2}^\infty\frac{x^2}{(k-2)!}=x^2e^x.$$ etc.