A survey of 80 sophomores at a certain western college showed that

36 take English
32 take history
32 take political science
16 take history and English
16 take political science and history
14 take political science and English
6 take all three

How many students:
1. take English and niether of the other two?
2. take none of the three courses?
3. take history, but niether of the other two
4. take political science and history but not english
5. do not take political science?

1. take English and niether of the other two?
36-6 that we know take all 3
This leave 30.
Subtract the 6 from those taking history and English.  Leaves 10.
Subtract the 10 from those taking English.
This leaves 20.
Subtract the 6 from those taking political science and English.  Leaves 8.
Subtract 8 from those taking English.
Leaves 12

2. take none of the three courses?

3. take history, but niether of the other two
Do the same with history as we did with English.
32-6 = 26 -10=16-10=6

4. take political science and history but not english
16-6 (that take all 3) = 10

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