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Question
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
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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.
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