In a village, there are 87 families of which 52 families have at most 2 children.

Question:

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

Solution:

52 families have at most 2 children, while 35 families have more than 2 children.

The selection of 20 families of which at least 18 families must have 2 children can be made in the ways given below.

(i) 18 families out of 52 and 2 families out of 35

(ii) 19 families out of 52 and 1 family out of 35

(iii) 20 families out of 52

$\therefore$ Required ways $={ }^{52} C_{18} \times{ }^{35} C_{2}+{ }^{52} C_{19} \times{ }^{35} C_{1}+{ }^{52} C_{20} \times{ }^{35} C_{0}$

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