In a small village, there are 87 families, of which 52 families have at most 2 children. In a rural development programme, 20 families are to be chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made?

Total number of families = 87

Number of families with at most 2 children = 52

Remaining families = 87 – 52 = 35

Also, for the rural development programme, 20 families are to be chosen for assistance, of which at least 18 families must have at most 2 children.

Thus, the following are the number of possible choices:

⁵²C₁₈ × ³⁵C₂ (18 families having at most 2 children and 2 selected from other types of families)

⁵²C₁₉ × ³⁵C₁ (19 families having at most 2 children and 1 selected from other types of families)

⁵²C₂₀ (All selected 20 families having at most 2 children)

Hence, the total number of possible choices = ⁵²C₁₈ × ³⁵C₂ + ⁵²C₁₉ × ³⁵C₁ + ⁵²C₂₀