1. [2] Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only 5 tickets remain, so 4 of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game, how many ways are there of choosing which 5 people can see the game?
Answer: 104 Let us count the number of way of distributing the tickets so that one of the conditions is violated. There is 1 way to give all the tickets to MIT students, and $\binom{7}{5}$ ways to give all the tickets to the 7 students other than Jack and Jill. Therefore, the total number of valid ways is $\binom{9}{5}-1-\binom{7}{5}=104$.