Question

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

• A. $^{52}C_{18} \,{}^{35}C_{2}$
• B. ${ }^{52} C _{18} \times{ }^{35} C _{2}+{ }^{52} C _{19} \times{ }^{35} C _{1}+{ }^{52} C _{20} \times{ }^{35} C _{0}$
• C. $^{52}C_{18}+\,{}^{35}C_{2} + \,{}^{52}C_{19} $
• D. $^{52}C_{18}\times \,{}^{35}C_{2} + \,{}^{35}C_{1} \times\,{}^{52}C_{19} $

Answer 1

Answer: (B)

The following are the number of possible choices:
$^{52}C_{18}\times \,{}^{35}C_{2} $ ( $18$ families having atmost $2$ children and $2$ selected from other type of families)
$^{52}C_{19} \times\,{}^{35}C_{1} $ ( $19$ families having at most $2$ children and $1$ selected from other type of families)
$^{52}C_{20}$ (All selected $20$ families having atmost $2$ children)
Hence, the total number of possible choices
$ =\,{}^{52}C_{18}\times\,{}^{35}C_{2} + \,{}^{52}C_{19} \times \,{}^{35}C_{1} +\,{}^{52}C_{20}$