# 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?

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 .

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