, 2017

Solution:

Consider a graph with five vertices corresponding to the roles, and draw an edge between two vertices if a player picks both roles. Thus there are exactly $5$ edges in the graph, and we want to find the probability that each vertex has degree $2$. In particular, we want to find the probability that the graph is composed entirely of cycles.

Thus there are two cases. The first case is when the graph is itself a $5$-cycle. There are $4!$ ways to choose such a directed cycle (pick an arbitrary vertex $A$ and consider a vertex it connects to, etc.), and thus $\frac{4!}{2} = 12$ ways for the undirected graph to be a $5$-cycle. Now, there are $5!$ ways to assign the edges in this cycle to people, giving a total contribution of $12 \cdot 5!$.

The second case is when the graph is composed of a $2$-cycle and a $3$-cycle, which only requires choosing the two vertices to be the $2$-cycle, and so there are $\binom{5}{2} = 10$ ways. To assign the players to edges, there are $\binom{5}{2} = 10$ ways to assign the players to the $2$-cycle. For the $3$-cycle, any of the $3! = 6$ permutations of the remaining $3$ players work. The total contribution is $10 \cdot 10 \cdot 6$.

Therefore, our answer is
$$\frac{12 \cdot 120 + 10 \cdot 10 \cdot 6}{10^5} = \frac{51}{2500}$$