105 students were asked if they liked soccer, rugby, or football. 4 liked only rugby, 40 liked rugby, 12 liked only rugby and football, 63 did not like football, 21 didn't like any, 16 liked only soccer and football, 60 liked soccer. How many liked only football? How many liked all three sports? How many liked football and rugby? How many liked football or rugby? I drew a Venn Diagram with 3 circles. When I label the regions without information with letters to solve for the information, I labeled the soccer only with a, the football only with b, the soccer and rugby intersection with c. How do I find the intersection of all 3? Should it be c+d or d-c or just d?
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To find the intersection of all three sports in this case, we can use the principle of inclusion-exclusion.
Let's label the regions in the Venn diagram as follows:
- "a" represents the region for soccer only
- "b" represents the region for football only
- "c" represents the region for soccer and rugby
- "d" represents the region for football and rugby
- "e" represents the region for rugby only
We need to determine the value of "d", which represents the intersection of all three sports.
To find the intersection of all three, we need to consider the total number of students who liked football (including those who liked only football and those who liked both football and rugby). According to the given information, 105 students were surveyed in total, and we know that 63 did not like football.
So, the number of students who liked football is 105 - 63 = 42.
From the given information, we also know that 12 students liked both rugby and football. Therefore, the number of students who liked only football can be found by subtracting the students who liked both rugby and football (d) from the total students who liked football (b + d).
42 = b + 12
b = 30
Now that we have found the value of b, which represents the number of students who liked only football, we can find the value of d, which represents the intersection of all three sports.
d = b - c
We don't have the value of c, which represents the region for soccer and rugby, in the given information. Therefore, we can't determine the exact value of d or the intersection of all three sports.
In summary, based on the information provided, we cannot determine the number of students who liked all three sports or the exact value of d.