In a certain group of 50 students at Hogwarts School of Wizardry, only 15 students are wearing caps and 30 students are wearing neckties. If 9 students are neither wearing a cap nor a necktie, how many students are wearing both a cap and a necktie?    3 4 5 6

The annual National No Spying Day is celebrated at KAOS headquarters this year. There are 11 Control agents and 19 KAOS agents attending. How many ways can we choose a team of 5 agents if 2 team members need to be from Control and 3 from KAOS?   How many ways can we choose a team of 5 agents if at least 1 team member needs to be from Control?

There is a 4-on-4 dodgeball game (8 players total). After the game, everyone shakes hands with everyone else once, including people on their team. 1. How many handshakes were there? 2. If it was a 5-on-5 dodgeball game, how many handshakes might there be? 3. How might you find out if there were 50 people on each team? Or 111? (You do not have to solve. Just explain or show how you would find it.)

The annual National No Spying Day is celebrated at KAOS headquarters this year. There are 11 Control agents and 14 KAOS agents attending. How many ways can we choose a team of 9 agents if 3 team members need to be from Control and 6 from KAOS? (Trivia note: the spying agencies are from Get Smart, a TV series from the late 1960's. I only watched it in reruns :-) --Dr. N.) How many ways can we choose a team of 9 agents if at least 1 team member needs to be from Control?