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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
avatar+124596 
+1

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

 

cool cool cool 

 Aug 8, 2022
 #2
avatar+2446 
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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
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+1

Thank you both!

 Aug 8, 2022

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