Determine How many player play foot ball?

In a group of athletic teams in a specific institute, 21 players are in the basket ball team, 26 players in the hockey team, 29 players in the foot ball team. If 14 players play hockey and basketball, 12 play foot ball and basket ball, 15 players play hockey and foot ball, 8 play all the three games.

   (i) How many players are there in all?

   (ii) How many play only foot ball?   (4+4)     

Ans: Suppose A, B and C is set of players who play Basket ball, Hockey and Foot ball correspondingly.

So

 |A| = 21,   |B| =26,   |C| = 29, 

|A∩B| = 14,   |A∩C| = 12,   |B∩C| = 15, and 

|A∩B∩C| = 8.   

Here now for part (i), we have to find out |A∪B∪C|. This can be calculated by using the formula   

|A∪B∪C| = |A| + |B| +  |C| - | A∩B| - |A∩C| - |B∩C|  + |A∩B∩C| 

  = 21 + 26 + 29 - 14 - 12 - 15 + 8 = 84 - 41 = 43.  

For part (ii) we have to find out number of players playing only football. This is   

  = |C| - |A∩C| - |B∩C| + |A∩B∩C| = 29 - 12 - 15 + 8 = 10.