# Thread: Combinatorics: How many students don't play any sports...

1. ## Combinatorics: How many students don't play any sports...

There are 27 students in a class. Of these 7 play basketball, 12 play hockey and 8 play soccer. In addition it is known that of those that play basketball 4 play hockey as well and 4 that play hockey play soccer as well. Both basketball and soccer is played by 3students. Only 2 students play all three sports. How many of the students in this group do not play any of the three sports?

I got the answer 12 but I'm not sure that it is correct. My solution is just to write everything out as below. It seems kind of inelegant and would be effiecient for higher numbers. Is there a better approach?

I first wrote out all the basketball players:

B, B, B, B, B, B, B

4 of these play hockey:

BH, BH, BH, BH, B, B, B

There are another 8 hockey players that can't also play basketball:

BH, BH, BH, BH, B, B, B, H, H, H, H, H, H, H, H

2 play all three:

BHS, BHS, BH, BH, B, B, B, H, H, H, H, H, H, H, H

Another 2 play hockey and soccer but not basketball:

BHS, BHS, BH, BH, B, B, B, HS, HS, H, H, H, H, H, H

3 play basketball and soccer:

BHS, BHS, BH, BH, BS, B, B, HS, HS, H, H, H, H, H, H

This gives 15 students that play a sport so 27-15=12 gives 12 that do not play any of the three sports.

Is this correct? Is there a better way to do it?

Thanks!

2. ## Re: Combinatorics: How many students don't play any sports...

Originally Posted by usersmuser
There are 27 students in a class. Of these 7 play basketball, 12 play hockey and 8 play soccer. In addition it is known that of those that play basketball 4 play hockey as well and 4 that play hockey play soccer as well. Both basketball and soccer is played by 3students. Only 2 students play all three sports. How many of the students in this group do not play any of the three sports?
How many out side the circles?