In a survey of 600 students in a school, 325 students were defence aspirants and 150 students were civil services aspirants, 100 students were both defence and civil services aspirants. Find how many students was neither defence nor civil services aspirant.

In a survey of 600 students in a school, 325 students were defence aspirants and 150 students were civil services aspirants, 100 students were both defence and civil services aspirants. Find how many students was neither defence nor civil services aspirant. Correct Answer 225

Concept:

Let A, B and C be three finite sets and U is the finite universal set, then

1. n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
2. n (A ∪ B) = n (A) + n (B) ⇔ A ∩ B = ϕ
3. n (A - B) = n (A) – n (A ∩ B) = n (A ∩ B’)
4. n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n(A ∩ B) – n (B ∩ C) – n (A ∩ C) + n (A ∩ B ∩ C)
5. n (A’ ∪ B’) = n = n (U) – n (A ∩ B)
6. n (A’ ∩ B’) = n = n (U) – n (A ∪ B)
7. n (A Δ B) = n (A) + n (B) – 2 n (A ∩ B)
8. n (A’) = n (U) – n (A)

Calculation:

Let, D = No. students who are defence aspirants and C = No. of students who are civil services aspirant.

Given: n (U) = 600, n (D) = 325, n (C) = 150 and n (D ∩ C) = 100.

As we know that, if A and B are finite aspirants then n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

⇒ n (D ∪ C) = n (D) + n (C) - n (D ∩ C)

⇒ n (D ∪ C) = 325 + 150 – 100

⇒ n (D ∪ C) = 375

So, the number of students who are neither defence nor civil services aspirants is given by: n

⇒ n = n (U) - n (D ∪ C) = 600 – 375 = 225.