In a survey of students about favorite sports, the results include 23 who like tennis, 28 who like football, 9 who like tennis and football, who like tennis and baseball, 16 who like football and baseball, 21 who like all three sports, and who like none 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?
To find the number of students who like only tennis and football, we need to subtract the number of students who like both tennis and football from the total number of students who like tennis.
1. Students who like only tennis and football:
- Count the number of students who like both tennis and football. In this case, it is given as 9 students who like both tennis and football.
2. To find the number of students who like only tennis and baseball, we need to subtract the number of students who like both tennis and baseball from the total number of students who like tennis.
- Count the number of students who like both tennis and baseball. In this case, it is not explicitly given.
- However, they mention that there are 21 students who like all three sports (tennis, football, and baseball). Since we know the total number of students who like tennis (23), and 21 of them like all three sports, we can calculate the number of students who like only tennis and baseball using the formula:
Number of students who like only tennis and baseball = Total number of students who like tennis - Number of students who like all three sports
3. To find the number of students who like only baseball and football, we need to subtract the number of students who like both baseball and football from the total number of students who like baseball.
- Count the number of students who like both baseball and football. In this case, it is given as 16 students who like both baseball and football.
Once you have these numbers, you can calculate the total number of students who like none of the sports using the formula:
Total number of students who like none of the sports = Total number of students - (Students who like only tennis + Students who like only football + Students who like only baseball + Students who like all three sports).
Note: The given data is incomplete. To find the exact number of students who like only tennis and baseball, we need additional information.
To solve this problem, we can use the principle of inclusion-exclusion.
Let's first find the total number of students who like either tennis, football, or baseball. From the given information, we have:
Number of students who like tennis: 23
Number of students who like football: 28
Number of students who like tennis and football: 9
Number of students who like tennis and baseball: ?
Number of students who like football and baseball: 16
Number of students who like all three sports: 21
Number of students who like none of the sports: ?
To find the number of students who like either tennis, football, or baseball, we can add the number of students who like each sport individually and then subtract the number of students who like both pairs of sports, so they are not counted twice.
Total number of students who like either tennis, football, or baseball:
= Number of students who like tennis + Number of students who like football + Number of students who like baseball
- Number of students who like both tennis and football - Number of students who like both tennis and baseball - Number of students who like both football and baseball
+ Number of students who like all three sports
= 23 + 28 + x - 9 - x - 16 + 21 (We don't know the values of x and the number of students who like none of the sports, so we'll replace them with x for now.)
Simplifying the equation:
Total number of students who like either tennis, football, or baseball:
= 23 + 28 + x - 9 - x - 16 + 21
= 47
We know that the total number of students is the sum of the number of students who like any of the three sports plus the number of students who like none of the sports:
Total number of students = Number of students who like either tennis, football, or baseball + Number of students who like none of the sports
= 47 + Number of students who like none of the sports
Since we know that the total number of students is not given, we'll represent it as a variable, T:
T = 47 + Number of students who like none of the sports
To find the number of students who like none of the sports, we'll set up an equation using the remaining information:
Number of students who like none of the sports + Number of students who like all three sports = Total number of students - Total number of students who like either tennis, football, or baseball
Number of students who like none of the sports + 21 = T - 47
Substituting the value of T we found earlier:
Number of students who like none of the sports + 21 = (47 + Number of students who like none of the sports) - 47
Number of students who like none of the sports + 21 = Number of students who like none of the sports
Simplifying the equation:
21 = 0
Oops! We have arrived at an incorrect equation. It appears that there is an error in the given information or a contradiction in our calculations. Please double-check the problem statement for accuracy or provide additional information if any part is missing.
To solve this problem, we can use the principle of Inclusion-Exclusion.
Let's define the variables:
- n(T) represents the number of students who like tennis.
- n(F) represents the number of students who like football.
- n(B) represents the number of students who like baseball.
- n(T∩F) represents the number of students who like both tennis and football.
- n(T∩B) represents the number of students who like both tennis and baseball.
- n(F∩B) represents the number of students who like both football and baseball.
- n(T∩F∩B) represents the number of students who like all three sports.
- n(Ω) represents the total number of students.
We are given the following information:
n(T) = 23
n(F) = 28
n(T∩F) = 9
n(T∩B) = ?
n(F∩B) = 16
n(T∩F∩B) = 21
n(Ω) = ? (unknown)
Using the principle of Inclusion-Exclusion, we can calculate the values of n(T∩B) and n(Ω).
n(Ω) = n(T) + n(F) + n(B) - n(T∩F) - n(T∩B) - n(F∩B) + n(T∩F∩B)
We can substitute the known values into the equation:
n(Ω) = 23 + 28 + n(B) - 9 - n(T∩B) - 16 + 21
Simplifying the equation gives:
n(Ω) = 47 + n(B) - n(T∩B) - 4
We also know that n(Ω) represents the total number of students, which means it must equal the sum of all the students who like each sport plus those who like none:
n(Ω) = n(T) + n(F) + n(B) - n(T∩F) - n(T∩B) - n(F∩B) + n(T∩F∩B) + n(None)
Since we are given that n(None) = 0, we can substitute it into the equation:
n(Ω) = 23 + 28 + n(B) - 9 - n(T∩B) - 16 + 21 + 0
Simplifying the equation gives:
n(Ω) = 47 + n(B) - n(T∩B) - 4
This equation is the same as the previous one for n(Ω), so we can equate them:
47 + n(B) - n(T∩B) - 4 = 47 + n(B) - n(T∩B) - 4
Both sides of the equation cancel out, so we are left with:
0 = 0
This means that the total number of students is the same as the sum of the students who like each sport plus those who like none. Since n(None) = 0, this equation holds true.
Therefore, n(Ω) = 47 + n(B) - n(T∩B) - 4
Now we can solve for n(T∩B) using the equation:
n(Ω) = 47 + n(B) - n(T∩B) - 4
Since n(Ω) is equal to the total number of students and we are not given that value, we cannot determine the exact number of students who like only tennis and football, only tennis and baseball, or only baseball and football.