1. Thirty people at a party shook hands with each other. How many handshakes were there altogether?
2. At a party, everyone shook hands with everybody else. There were 45 handshakes. How many people were at the party?
3. Every student in the second-grade classroom exchanged a valentine card with each other. If there were 30 students, how many valentine cards were exchanged?
4. Twenty people are sitting in a circle. Each person shakes hands with everyone but his/her neighbors. How many handshakes have been exchanged?
5. Two ten-member volleyball teams play a game. After the game, each of the members of the winning team shakes hands once with each member of both teams. How many handshakes take place?
6. Eight students of different heights are at a party. Each student decides to only participate in a handshake with another student shorter than himself or herself. How many handshakes take place?
7. Each person at a graduation party shook hands with everyone else. John shook hands with three times as many men as women. John’s wife shook hands with four times as many men as women. How many men and women were there at the party?
8. Find the number of diagonals of a 12-sided polygon?
9. If you build a staircase with cubes that is 10 steps high, how many cubes will you need?
10. At a party, each man shook hands with everyone except his spouse, and no handshakes took place between women. If 13 married couples attended, how many handshakes were there among these 26 people?
(A) 78 (B) 185 (C) 234 (D) 312 (E) 325
Figure 1.12 Using technology to represent the worked examples of the handshake problem.
11. (AMC8 2012) In the BIG N, a middle-school football conference, each team plays
every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10
12. (AMC12 2011) At a twins and triplets convention, there were nine sets of twins and six sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?