SOLUTION: In a group of 28 students, 11 are taking math, 19 are taking English, 12 are taking history, 6 are taking math and history, 9 are taking history and English, 7 are taking math and

Question 255058: In a group of 28 students, 11 are taking math, 19 are taking English, 12 are taking history, 6 are taking math and history, 9 are taking history and English, 7 are taking math and English, and 5 are taking all three subjects.
(a) How many of the students are not taking any of the three subjects?
(b) How many of the students are taking only math?
(c) How many of the students are taking only one of these subjects?

Answer by Greenfinch(383) (Show Source): You can put this solution on YOUR website!
This is clearer if you do a Venn diagram. Anyway, here are the numbers.
M + H + E = 5
so M + H = 6 and therefore 1 is just doing M + H
and M + E = 7 therefore 2 are just doing M + E
and E + H = 9 therefore 4 are just doing E + H
11 are doing M from which you subtract the 5 doing all, the 1 doing M + H and the 2 doing M + E giving 3 just doing M
12 are doing H from which you subtract the 5 doing all, the 1 doing M + H and the 4 doing E + H giving 2 just doing H
19 are doing E, from which you subtract the 5 doing all, the 4 doing E + H and the 2 doing M +E giving 8 just doing English.
To summarise therefore
M+E+H = 5
M+E = 2
M+H = 1
E + H = 4
M = 3
E = 8
H = 2
which totals 25, so 3 are doing nothing. 3 are doing only M, and 3 + 2 + 8 = 13 are doing one subject.
If you can transcribe these to a Venn diagram it will look clearer
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