23 Jul 3qDiscrete
1. A rook is a chess piece that can attack another piece if they are in the same row or the same column. A chess board has 8 rows and 8 columns.
(a) How many ways can you place 8 rooks so that no two can attack each other? We assume that the rooks are identical so interchanging two rooks does not count as a separate placement.
(b) Suppose that you have 4 wooden and 4 marble rooks. How many ways can you do this?
(c) How many ways can you do this if all the 8 rooks are di§erent?
2. How many ways can we give 25 (identical) pencils to ten children so that each child will receive at least one?
3. A college football coach must choose four new players who will receive scholarships. He can choose among 20 incoming players, half of whom are o§ensive players, and half of whom are defensive players.
(a) How many ways can the coach award the scholarships?
(b) If there must be at least one o§ensive player and at least one defensive player receiving the scholarship, how many ways can the coach award the scholarships?
4. (9.7 #34) Find the coe¢cient of x9y10 in (2x− 3y2)14 .
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