Out of 1865 people, 660 can speak English and 1305 can speak Hindi, but 120…

Concept

For two sets E and H, inclusion–exclusion gives |E ∪ H| = |E| + |H| − |E ∩ H|.

The union counts everyone in at least one set, while the intersection is subtracted once because it was counted in both individual totals.

Application

1. People speaking at least one language: |E ∪ H| = 1865 − 120 = 1745.

2. Let x = |E ∩ H|, the number speaking both languages.

3. Substitute the data: 1745 = 660 + 1305 − x.

4. Combine the two language totals: 1745 = 1965 − x.

5. Rearrange: x = 1965 − 1745 = 220.

Cross-check

1. English only = 660 − 220 = 440.

2. Hindi only = 1305 − 220 = 1085.

3. The disjoint language groups total 440 + 220 + 1085 = 1745; adding the 120 people who speak neither gives 1865.

Therefore, 220 persons can speak both languages.