Answer Happy

2. (15 points) A workplace has 12 staff, and Bob and Dan are among them. (a) How many teams of 5 can be formed? (b) How many teams of 5 are there if both Bob and Dan must be in each team. (c) Now Bob and Dan insist on working as a pair (a team must contain either both or neither). How many teams of 5 are there?

3. (15 points) A soccer team has 12 men and 5 women players. They all show up for a game. Only 11 players can take the field. (a) How many ways are there to choose the players who take the field? (b) Each player in the team can play any of the 11 positions (goalkeeper, striker, ...). How many ways are there to assign the 11 positions by selecting players from those who show up? (c) Now, at least one player must be a woman. How many ways are to choose the 11 players?

4. (10 points) Give a combinatorial proof of each of the following identity. (a) By using the cardinality of the power set, show that {k=0 (z) = 2". (b) By counting the number of committees of 3 such that one is a leader, show that n("71) = (2)(n − 2).