The number of students that like all three sports is included in the number of students that like any combination of sports, such as tennis and football.
Only Tennis and Football: 3 Only Tennis and Baseball: 11 Only Baseball and Football: 14
Practice makes perfect
Let's first draw a Venn diagram representing the given situation.
From the survey, we know the following information about the students' favorite sports.
r|c
Tennis & 22
Football& 25
Football and Tennis& 9
Tennis and Baseball& 17
Football and Baseball& 20
All Sports& 6
No Sports& 4
Using our Venn diagram, we can illustrate the outcome of the survey. From the results, we know that 6 students like all three sports. This will be written in the intersection of all three circles. We also know that 4 students like none of the sports. We write this number outside of the circles.
From the survey, we know that 9 students like both tennis and football. This number includes the students that like all three sports. Therefore, the number of students that only like tennis and football must be 9-6=3.
Similarly, 17 students like both tennis and baseball. This number includes the students that like all three sports. Therefore, the number of students that only like tennis and baseball must be 17-6=11.
Finally, we know that 20 students like football and baseball. This means that 14 students like only football and baseball.
Let's summarize what we have found.
r|c
Only Football and Tennis& 3
Only Tennis and Baseball& 11
Only Football and Baseball& 14
