in a survey of students about their favorite sports, the results include 25 who like tennis, 31 who like football, 10 who like tennis and football, 18 who like tennis and baseball, 24 who like football and baseball, 5 who like all three, and 9 who like none of the sports. how many students like only tennis and football? how many like only tennis and baseball? how many students like only baseball and football?
To solve this problem, we can use a Venn diagram. Let's first label the overlapping regions:
- Let's call the region where tennis and football overlap "Tennis and Football".
- Let's call the region where tennis and baseball overlap "Tennis and Baseball".
- Let's call the region where football and baseball overlap "Football and Baseball".
We are given the following information:
- There are 25 students who like tennis.
- There are 31 students who like football.
- There are 10 students who like both tennis and football.
- There are 18 students who like both tennis and baseball.
- There are 24 students who like both football and baseball.
- There are 5 students who like all three sports.
- There are 9 students who like none of the sports.
Using this information, we can fill in the Venn diagram as follows:
- The number of students who like only tennis can be calculated as (25 - 10 - 18 - 5), which is 25 - 33 = -8. Since we cannot have a negative number of students, we assume this value is 0 (i.e., there are no students who like only tennis).
- The number of students who like only football can be calculated as (31 - 10 - 24 - 5), which is 31 - 39 = -8. Again, we assume this value is 0 (i.e., there are no students who like only football).
- The number of students who like only baseball can be calculated as (24 - 18 - 5), which is 24 - 23 = 1.
- The number of students who like tennis and football can be calculated as (10 - 5), which is 10 - 5 = 5.
- The number of students who like tennis and baseball can be calculated as (18 - 5), which is 18 - 5 = 13.
- The number of students who like football and baseball can be calculated as (24 - 5), which is 24 - 5 = 19.
Finally, let's calculate the number of students who like only tennis and football:
- This is the number of students who like tennis and football (5) minus the number of students who like all three sports (5), which is 5 - 5 = 0.
Therefore, there are no students who like only tennis and football.
The number of students who like only tennis and baseball is 13.
The number of students who like only baseball and football is 1.
To solve this problem, we can use a method called the principle of inclusion-exclusion.
First, let's break down the given information:
- The number of students who like tennis: 25
- The number of students who like football: 31
- The number of students who like tennis and football: 10
- The number of students who like tennis and baseball: 18
- The number of students who like football and baseball: 24
- The number of students who like all three sports: 5
- The number of students who like none of the sports: 9
Now, let's calculate the number of students who like only tennis and football. To do this, we need to subtract the number of students who like tennis and football from the total number of students who like tennis:
Number of students who like only tennis and football = Number of students who like tennis - Number of students who like tennis and football
Number of students who like only tennis and football = 25 - 10 = 15
Similarly, to find the number of students who like only tennis and baseball, we subtract the number of students who like tennis and baseball from the total number of students who like tennis:
Number of students who like only tennis and baseball = Number of students who like tennis - Number of students who like tennis and baseball
Number of students who like only tennis and baseball = 25 - 18 = 7
To find the number of students who like only baseball and football, we need to subtract the number of students who like all three sports and the number of students who like tennis and baseball from the total number of students who like football:
Number of students who like only baseball and football = Number of students who like football - Number of students who like football and baseball - Number of students who like all three sports
Number of students who like only baseball and football = 31 - 24 - 5 = 2
Therefore, the number of students who like only tennis and football is 15, the number of students who like only tennis and baseball is 7, and the number of students who like only baseball and football is 2.
To answer these questions, we can use a method called the Principle of Inclusion-Exclusion. Let's break it down step-by-step:
Step 1: Calculate the total number of students who like tennis, football, or baseball:
- Students who like tennis = 25
- Students who like football = 31
- Students who like baseball = Total students who like tennis or baseball - Students who like all three - Students who like neither = 25 + 18 - 5 - 9 = 29
Step 2: Calculate the number of students who like both tennis and football:
- Students who like tennis and football = 10
Step 3: Calculate the number of students who like both tennis and baseball:
- Students who like tennis and baseball = 18
Step 4: Calculate the number of students who like both baseball and football:
- Students who like baseball and football = Total students who like baseball or football - Students who like all three = 24
Step 5: Calculate the number of students who like only tennis and football:
- Students who like only tennis and football = Students who like tennis and football - Students who like all three - Students who like both tennis and baseball = 10 - 5 - 18 = -13
- Since a negative number doesn't make sense in this context, we can conclude that there are 0 students who like only tennis and football.
Step 6: Calculate the number of students who like only tennis and baseball:
- Students who like only tennis and baseball = Students who like tennis and baseball - Students who like all three - Students who like both tennis and football = 18 - 5 - 10 = 3
Step 7: Calculate the number of students who like only baseball and football:
- Students who like only baseball and football = Students who like baseball and football - Students who like all three = 24 - 5 = 19
So, the final answers are:
- Number of students who like only tennis and football = 0
- Number of students who like only tennis and baseball = 3
- Number of students who like only baseball and football = 19