Answer Happy

You are a sneaker enthusiast and have bought four pairs of new shoes from Nike. However, Nike decides to send each shoe in an individual box. (a) If you open four boxes at random, what is the probability that you do not have a pair that go together? (b) If you open five boxes, what is the probability that you have at most two pairs? A group of ten people get together to play a game of basketball at Barton Hall. Seven of the people are undergraduates and three of them are Professors. They will divide themselves into two teams of five players each. (a) In how many ways can this be done? (b) How many ways can it be done if each team needs at least one Professor?