In a survey of students about favorite sports, the results include 29 who like tennis, 31 who like football, 11 who like tennis and football, 19 who like tennis and baseball, 22 who like football and baseball, 4 who like all three sports, and 3 who like of the sports. How many students like only tennis and football? How many students like only tennis and baseball? How many students like only baseball and football? How many students like only tennis and football?
To solve this problem, we can use the principle of inclusion-exclusion and create a Venn diagram.
Let's denote:
A: students who like tennis
B: students who like football
C: students who like baseball
According to the given information:
A = 29
B = 31
A ∩ B = 11
A ∩ C = 19
B ∩ C = 22
A ∩ B ∩ C = 4
None of the sports = 3
We want to find the number of students who like only tennis and football, which is equal to:
(A ∩ B) - (A ∩ B ∩ C)
(A ∩ B) = 11
(A ∩ B ∩ C) = 4
Therefore, the number of students who like only tennis and football is 11 - 4 = <<11-4=7>>7.
Similarly, we can find the number of students who like only tennis and baseball:
(A ∩ C) - (A ∩ B ∩ C) = 19 - 4 = <<19-4=15>>15.
And the number of students who like only baseball and football:
(B ∩ C) - (A ∩ B ∩ C) = 22 - 4 = <<22-4=18>>18.
Lastly, we want to find the number of students who like only tennis and football:
A - (A ∩ B) - (A ∩ C) + (A ∩ B ∩ C) = 29 - 11 - 19 + 4 = <<29-11-19+4=3>>3.
Therefore, the number of students who like only tennis and football is 3.
To find the number of students who like only tennis and football, we need to subtract the students who like both tennis and football from the total number of students who like tennis and football.
Total students who like tennis and football = 11
Total students who like all three sports = 4
Therefore, the number of students who like only tennis and football = Total students who like tennis and football - Total students who like all three sports
= 11 - 4
= 7
Similarly, to find the number of students who like only tennis and baseball, we subtract the students who like both tennis and baseball from the total number of students who like tennis and baseball.
Total students who like tennis and baseball = 19
Total students who like all three sports = 4
Therefore, the number of students who like only tennis and baseball = Total students who like tennis and baseball - Total students who like all three sports
= 19 - 4
= 15
To find the number of students who like only baseball and football, we subtract the students who like both football and baseball from the total number of students who like football and baseball.
Total students who like football and baseball = 22
Total students who like all three sports = 4
Therefore, the number of students who like only baseball and football = Total students who like football and baseball - Total students who like all three sports
= 22 - 4
= 18
Finally, to find the number of students who like only tennis and football, we subtract the students who like all three sports from the total number of students who like tennis, football, and baseball.
Total students who like all three sports = 4
Therefore, the number of students who like only tennis, football, and baseball = Total students who like tennis and football - Students who like all three sports
= 11 - 4
= 7
To find out how many students like only tennis and football, tennis and baseball, baseball and football, and only tennis and baseball, we can use a concept called the inclusion-exclusion principle.
The inclusion-exclusion principle states that if you want to find the cardinality (number of elements) in a union of several sets, you can recursively add the sizes of individual sets, subtract the sizes of intersection of pairs of sets, add the sizes of the intersection of triplets of sets, and so on.
Let's define the sets:
T = set of students who like tennis
F = set of students who like football
B = set of students who like baseball
From the given information, we know:
|T| = 29 (number of students who like tennis)
|F| = 31 (number of students who like football)
|B| = ? (number of students who like baseball is not given)
We are also given:
|T ∩ F| = 11 (number of students who like both tennis and football)
|T ∩ B| = 19 (number of students who like both tennis and baseball)
|F ∩ B| = 22 (number of students who like both football and baseball)
|T ∩ F ∩ B| = 4 (number of students who like all three sports)
|T ∪ F ∪ B| = ? (number of students who like at least one sport is not given)
To find the numbers of students who like only tennis and football, only tennis and baseball, only baseball and football, and only tennis, we need to find the cardinalities of these sets.
1. Only tennis and football (|T - (T ∩ F)|):
This can be found by subtracting the number of students who like both tennis and football from the total number of students who like tennis: |T| - |T ∩ F|. Substituting the given values, we have: 29 - 11 = 18 students.
2. Only tennis and baseball (|T - (T ∩ B)|):
This can be found by subtracting the number of students who like both tennis and baseball from the total number of students who like tennis: |T| - |T ∩ B|. Substituting the given values, we have: 29 - 19 = 10 students.
3. Only baseball and football (|B - (F ∩ B)|):
This can be found by subtracting the number of students who like both football and baseball from the total number of students who like baseball: |B| - |F ∩ B|. However, we don't know the value of |B|, so we can't calculate this directly. It is missing information.
4. Only tennis:
To find the number of students who like only tennis, we can use the inclusion-exclusion formula. This is given by the equation:
|T - (T ∩ F) - (T ∩ B) - (T ∩ F ∩ B)|.
Substituting the given values, we have:
|T - (T ∩ F) - (T ∩ B) - (T ∩ F ∩ B)| = |T| - |T ∩ F| - |T ∩ B| + |T ∩ F ∩ B|.
Substituting the given values, we have:
29 - 11 - 19 + 4 = 3 students.
Therefore, the answers to your questions are:
- The number of students who like only tennis and football is 18.
- The number of students who like only tennis and baseball is 10.
- The number of students who like only baseball and football cannot be determined due to missing information.
- The number of students who like only tennis is 3.