In a group of 15, 7 have studied Latin. 8 have studied Greek and 3 have not studied either. How many of these studied both Latin and Greek?

Question: In a group of 15, 7 have studied Latin. 8 have studied Greek and 3 have not studied either. How many of these studied both Latin and Greek?

Solution:
Given that,
Total people = 15
Studied Latin, n(L) = 7
Studied Greek, n(G) = 8
Studied neither = 3

Therefore, the number who studied at least one of the two languages is, n(L ∪ G) = 15 - 3 = 12

We know,
n(L ∪ G) = n(L) + n(G) - n(L ∩ G)
⇒ 12 = 7 + 8 - n(L ∩ G)
⇒ 12 = 15 - n(L ∩ G)
∴ n(L ∩ G) = 3

Therefore, 3 people studied both Latin and Greek.