Question
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 ?
A.
$$^{52}{C_{18}} \times {\,^{35}}{C_2}$$
B.
$$^{52}{C_{18}} \times {\,^{35}}{C_2} + {\,^{52}}{C_{19}} \times {\,^{35}}{C_1} + {\,^{52}}{C_{20}}$$
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 :
$$^{52}{C_{18}} \times {\,^{35}}{C_2} + {\,^{52}}{C_{19}} \times {\,^{35}}{C_1} + {\,^{52}}{C_{20}}$$
Solution :
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 atmost 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 is :
$$ = {\,^{52}}{C_{18}} \times {\,^{35}}{C_2} + {\,^{52}}{C_{19}} \times {\,^{35}}{C_1}\, + {\,^{52}}{C_{20}}$$