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Balvinder
A group of 8 friends want to play doubles tennis. How many different ways can the group be divided into 4 teams of 2 people?

A. 420
B. 2520
C. 168
D. 90
E. 105

Official Solution


Credit: Veritas Prep

It is quite clear here that the teams are not distinct i.e. we don’t have team 1, team 2 etc. But let’s solve this question by first making team 1, team 2, team 3 and team 4. Later we will adjust the answer.

Out of 8 people, in how many ways can we make team 1? In 8*7/2! ways (i.e. 8C2).
Out of 6 people, in how many ways can we make team 2? In 6*5/2! ways (i.e. 6C2).
Out of 4 people, in how many ways can we make team 3? In 4*3/2! ways (i.e. 4C2).
Out of 2 people, in how many ways can we make team 4? In 2*1/2! ways (i.e. 2C2).

In how many ways can we make the 4 teams? In 8*7*6*5*4*3*2*1/(2!*2!*2!*2!) = 8!/(2!*2!*2!*2!) ways. But here, we have considered the 4 teams to be distinct. Since the teams are not distinct, we will just divide by 4!

We get 8!/(2!*2!*2!*2!*4!) = 105

ANSWER: E
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Visualize it this way -

Players = A,B,C,D,E,F,G,H

A can team up any of the 7 players and out of the remaining 6 players, we can select any 2 make a team (6C2)

Total ways = 7* 6C2 = 105 ways
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