Combinations Question (Please provide an explanation!)

Question

1. There are 140 students in the entire 12th Grade class.
2. 52 signed up for biology.
3. 71 signed up for chemistry.
4. 40 signed up for physics.
5. 15 signed up for biology and chemistry
6. 8 signed up for chemistry and physics.
7. 11 signed up for biology and physics.
8. 2 signed up for all 3 sciences.

a) How many students are NOT taking any science?

b) Illustrate this using a Venn diagram.

Please provide an explanation, I don't understand.

Answer 1

Okay, this is where drawing the diagram helps me make it make sense.

Draw the three overlapping circles. Add the labels and numbers given.

Biology circle has 52 for the whole circle, the intersection with Chemistry has 15 (which includes the 2 who take all three sciences), the intersection with Physics has 11 ( which also includes the 2 in all three). So, doing the subtraction you end up with

Biology-Chemistry 15 - 2 = 13

(Bio-Chem total - All three)

Biology-Physics 11 - 2 = 9

(Bio-Phy total - all three)

Biology-Chemistry-Physics = 2

(students taking all three)

Biology alone 52 - 13 - 9 - 2 = 28

(Bio total - Bio-Chem - Bio-Phy - all three)

Now Chemistry

Biology-Chemistry already accounted for (13)

Chemistry-Physics 8 - 2 = 6

(Chem-Phy total - all three)

Chemistry alone 71 - 13 - 6 - 2 = 50

(Chem total - Bio-Chem - Chem-Phy - all three)

Last is Physics

Chemistry Physics already accounted for (6)

Biology-Physics already accounted for (9)

Physics alone 40 - 6 - 9 - 2 = 23

(Phy total - Chem-Phy - Bio-Phy - all three)

Now add them all

Bio alone + Chem alone + Phy alone + Bio-Chem + Bio-Phy + Chem-Phy + all three sciences

28 + 50 + 23 + 13 + 9 + 6 + 2 =131

Now take all of the students and subtract the ones taking sciences

140 - 131 = 9

Answer 2

7 took no science course

draw three overlapping circles, one for biology, one for chemistry and one for physics

all 3 overlap in the middle with 2 students taking all 3 courses

15-2 = 13 who take both biology & chemistry

8-2 = 6 who take both chemistry & physics

11-2 - 9 who take both biology and physics

the overlaps sum to 30. 13+9+6+2

52+40+71 = 163

subtract 30 from 163 = 133. 140-133 = 7 students who took no science course