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?

Question

Philoid · Accepted Answer

Given that 52 families out of 82 families have at most 2 children.

It is told that 20 families need to be selected with at least 18 families having utmost 2 children.

The following are the cases:

i. 18 families having at most 2 children and 2 from other families

ii. 19 families are having at most 2 children and 1 from other families

iii. 20 families having at most 2 children

Let us assume that the no. of ways choosing to be N.

⇒ N = (Selecting 18 families having at most 2 children and 2 from other families) + (Selecting 19 families having at most 2 children and 1 from other families) + (Selecting 20 families having at most 2 children)

⇒ N = (⁵²C₁₈ × ³⁵C₂) + (⁵²C₁₉ × ³⁵C₁) + (⁵²C₂₀)

We know that ,

And also n! = (n)(n – 1)......2.1

⇒

∴ The no. of ways of choosing 18 families are .