A wedding reception has exactly 2n guests. There are n children and n adults. Each guest writes the

name of their preferred dance partners on a piece of paper (allowed more than one!). A couple will dance

together if and only if they each chose the other person as their preferred dance partner. For example, if

Carol wants to dance with Bob and Bob wants to dance with Carol, then they will dance. Suppose children

like to chance with children, so the chance that any child selects another child as a dance parter is 1=2. A

child selects any adult as a dance partner with probability 1=10. On the other hand, adults like to dance

with adults. The chance that any adult selects another adult as a dance partner is 1=4. The chance that

an adult selects a child as a dance parter is only 1=20. Determine the expected number of dances that

take place at the reception. Next, repeat the process assuming each guest is only allowed to select exactly

one partner. Children will select among all guests equally, and adults will select an adult twice as often

as they select a child.