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# Counting Question

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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.

Aug 8, 2022

#1
+128826
+2

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

.

.

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

Aug 8, 2022
#2
+2667
0

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 \(190 - 10 = \color{brown}\boxed{180}\)

Aug 8, 2022
#3
+1

Thank you both!

Aug 8, 2022