# Thread: Combinatorics: Upper and lower bounds

1. ## Combinatorics: Upper and lower bounds

In a class of students 21 play basketball, 32 students pay hockey, and 32 students play soccer. In addition we know that 12 students play both basketball and hockey, 14 students pay both basketball and soccer, 11 both hockey and soccer. Each student in this class plays at least one of the three sports. What is the smallest possible upper bound and the largest possible lower bound that can be determined with the given information for the number of students in this class?

Any hints on how to approach this?

2. ## Re: Combinatorics: Upper and lower bounds

Originally Posted by usersmuser
In a class of students 21 play basketball, 32 students pay hockey, and 32 students play soccer. In addition we know that 12 students play both basketball and hockey, 14 students pay both basketball and soccer, 11 both hockey and soccer. Each student in this class plays at least one of the three sports. What is the smallest possible upper bound and the largest possible lower bound that can be determined with the given information for the number of students in this class?
Any hints on how to approach this?
See your other post. Learn to draw Venn diagrams.

3. ## Re: Combinatorics: Upper and lower bounds

Originally Posted by Plato
See your other post. Learn to draw Venn diagrams.
I can't see how that works in this case where we have incomplete information?

4. ## Re: Combinatorics: Upper and lower bounds

Originally Posted by usersmuser
I can't see how that works in this case where we have incomplete information?
The lower or upper bound occurs when a maximum number of students play all 3 sports. You cannot have more than 11 students play all 3 sports. That's because only 11 students play both hockey and soccer. Then you have the opposite bound when zero students play all three sports. As Plato suggested, do the Venn diagrams.

5. ## Re: Combinatorics: Upper and lower bounds

Originally Posted by SlipEternal
The lower or upper bound occurs when a maximum number of students play all 3 sports. You cannot have more than 11 students play all 3 sports. That's because only 11 students play both hockey and soccer. Then you have the opposite bound when zero students play all three sports. As Plato suggested, do the Venn diagrams.
Oops, you need a minimum of 5 students that play all 3 sports because you have 21 basketball players, 12 that play basketball and hockey and 14 that play basketball and soccer. So, you need at least 5 players that play all three sports.