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Question
In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:
French = 17, English = 13, Sanskrit = 15 French and English = 09, English and Sanskrit = 4 French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study at least one of the three languages
Solution
Let us use Venn diagram method.
Total number of students = 50
⇒ n(U) = 50
Number of students who study French = 17
⇒ n(F) = 17
Number of students who study English = 13
⇒ n(E) = 13
Number of students who study Sanskrit = 15
⇒ n(S) = 15
Number of students who study French and English = 9
⇒ n(F ∩ E) = 9
Number of students who study English and Sanskrit = 4
⇒ n(E ∩ S) = 4
Number of students who study French and Sanskrit = 5
⇒ n(F ∩ S) = 5
Number of students who study French, English and Sanskrit = 3
⇒ n(F ∩ E ∩ S) = 3
n(F) = 17
a + b + d + e = 17 ......(i)
n(E) = 13
b + c + e + f = 13 ......(ii)
n(S) = 15
d + e + f + g = 15 ......(iii)
n(F ∩ E) = 9
∴ b + e = 9 ......(iv)
n(E ∩ S) = 4
∴ e + f = 4 .......(v)
n(F ∩ S) = 5
∴ d + e = 5 ......(vi)
n(E ∩ F ∩ S) = 3
∴ e = 3 .......(vii)
From (iv)
b + 3 = 9
⇒ b = 9 – 3 = 6
From (v)
3 + f = 4
⇒ f = 4 – 3 = 1
From (vi)
d + 3 = 5
⇒ d = 5 – 3 = 2
Now from equation (i)
a + 6 + 2 + 3 = 17
⇒ a = 17 – 11
⇒ a = 6
Now from equation (ii)
6 + c + 3 + 1 = 13
⇒ c = 13 – 10
⇒ c = 3
From equation (iii)
2 + 3 + 1 + g = 15
⇒ g = 15 – 6
⇒ g = 9
Number of students who study at least one of the three languages
= a + b + c + d + e + f + g
= 6 + 6 + 3 + 2 + 3 + 1 + 9
= 30
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