Out of 100 students; 15 passed in English, 12 passed in Mathematics, 8 in Science, 6 in English and Mathematics, 7 in Mathematics and Science; 4 in English and Science; 4 in all the three. Find how many passed in more than one subject only

#### Solution

Let the number of students passed in Mathematics M, E be in English and S be in Science.

Then n(U) = 100

n(M) = 12

n(E) = 15

n(S) = 8

n(E ∩ M) = 6

n(M ∩ S) = 7

n(E ∩ S) = 4

And n(E ∩ M ∩ S) = 4

Let us draw a Venn diagram.

According to the Venn diagram, we get

a + b + d + e = 15 ......(i)

b + c + e + f = 12 ......(ii)

d + e + f + g = 8 .....(iii)

n(E ∩ M) = 6

∴ b + e = 6 ......(iv)

n(M ∩ S) = 7

∴ e + f = 7 ......(v)

n(E ∩ S) = 4

∴ d + e = 4 ......(vi)

And n(E ∩ M ∩ S) = 4

∴ e = 4 ......(vii)

From equation (iv) and (vii)

We get b + 4 = 6

∴ b = 2

From equation (v) and (vii)

We get 4 + f = 7

∴ f = 3

From equation (vi) and (vii)

We get d + 4 = 4

∴ d = 0

From equation (i) we get

a + b + d + e = 15

⇒ a + 2 + 0 + 4 = 15

⇒ a = 9

From equation (ii)

b + c + e + f = 12

⇒ 2 + c + 4 + 3 = 12

⇒ c = 3

From equation (iii)

d + e + f + g = 8

⇒ 0 + 4 + 3 + g = 8

⇒ g = 1

∴ Number of students who passed in more than one subject = b + e + d + f = 2 + 4 + 0 + 3 = 9.