I don't know how to answer the following, so here goes.
10 people decide to exchange gifts, in the following manner:
All 10 people write down their names on pieces of paper, and along with their names go their gift preferences.
They put the names in a box and then each person randomly picks out a name from the box.
They then give their gifts to the person whose name they drew from the box. If a person draws his own name, then he gives his gift to himself.
Now, if Anne gives his gift to Bob, and Bob gives his gift to Charles, and Charles gives his gift to Anne, then Anne, Bob and Charles are said to be in a chain, with 3 members.
Now for the million-dollar question:
What is the expected length of the longest chain in the group?