10 teams containing 2 players each compete in a doubles tennis tournament. After the medal ceremony, every player shakes hands once with every other player except the other member of their team. How many handshakes occur?
So I thought it would be 19*20=380 because there are 20 players and each player shakes 19 hands but that is not correct.
The first team will have a total of 18 + 18 = 2*18 unique handshakes
The second team will have a total of 16 + 16 = 2*16 unique handshakes
The third team will have a total of 14 + 14 = 2*14 unique handshakes
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The ninth team has a total of 2 + 2 = 2 * 2 unique handshakes
So the total = 2 ( sum of the first 9 even positive integers) =
2 (n) (n + 1) =
2 (9) (10) =
180 handshakes
There are \({20 \choose 2} = 190\) ways to choose 2 people to shake hands.
Of these, there are 10 ways to shake hands with the same team member, so there are \(190 - 10 = \color{brown}\boxed{180}\)