The summary of a survey of 500 students listed the following totals:
* 259 students did math homework
* 219 students did English homework
* 238 students did science homework
* 108 students did English and science homework
* 81 students did science and math homework
* 85 students did math and English homework
* 40 students did no homework
How many students did math, science, and English homework?
To find the number of students who did math, science, and English homework, we need to find the intersection of all three sets: math, science, and English.
Let's break down the given information:
- 259 students did math homework.
- 219 students did English homework.
- 238 students did science homework.
- 108 students did English and science homework.
- 81 students did science and math homework.
- 85 students did math and English homework.
- 40 students did no homework.
To find the number of students who did math, science, and English homework, we can add up the numbers of students who did each combination and find the intersection:
108 (English and science) + 81 (science and math) + 85 (math and English) = 274.
So, 274 students did math, science, and English homework.
To find the number of students who did math, science, and English homework, we need to find the intersection of all three sets.
We can use the principle of inclusion-exclusion to solve this.
First, let's find the number of students who did math or science or English homework:
Total students who did math = 259
Total students who did science = 238
Total students who did English = 219
Adding these three totals together, we have: 259 + 238 + 219 = 716
But by doing so, we have counted some students multiple times because they did more than one subject's homework. So, we need to subtract the intersections of the sets.
Total students who did math and science = 81
Total students who did science and English = 108
Total students who did math and English = 85
Adding these three intersections together, we have: 81 + 108 + 85 = 274
By subtracting this value from the total number of students who did at least one subject's homework, we can find the number of students who did math, science, and English homework:
716 - 274 = 442
Therefore, 442 students did math, science, and English homework.