In a class of 50 students, 35 opted for mathematics and 37 opted for Biology. How many have opted for both Mathematics and Biology? How many have opted for only Mathematics? (Assume that each student has to opt for at least one of the subjects).

Given in the question,
n(M ∪ B) = 50, n(M) = 35, n(B) = 37, n(M ∩ B) =?
use the below formula
n(M ∪ B) = n(M) + n(B) – n(M ∩ B)
We get 50 = 35 + 37 – n(M ∩ B)
⇒ n(M ∩ B) = 35 + 37 – 50 = 72 – 50 = 22
∴ 22 students have opted for both Mathematics and Biology.
Again number of students who have opted for only Mathematics = n(M) – n(M ∩ B) = 35 – 22 = 13.