In a group of students, 100 students know Spanish, 50 know Arabic and 25 know both. Each of the students know either Spanish or Arabic. How many students are there in the group.

In a group of students, 100 students know Spanish, 50 know Arabic and 25 know both. Each of the students know either Spanish or Arabic. How many students are there in the group. Correct Answer 125

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, S = Number of students who speak Spanish and A = Number of students who can speak Arabic.

Let us assume there are x number of people in group.

Given: n (S) = 100, n (A) = 50 and n (S ∩ A) = 25.

∵ all the students know either Spanish or Arabic ⇒ n (S ∪ A) = x.

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

⇒ n (S ∪ A) = n (S) + n (A) – n (S ∩ A)

⇒ x = 100 + 50 – 25 = 125

Hence, there are 125 students in the group.