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?

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) 52C18 35C2 (B)  52 C 18 × 35 C 2+ 52 C 19 × 35 C 1+ 52 C 20 × 35 C 0 (C) 52C18+ 35C2 + 52C19  (D) 52C18× 35C2 + 35C1 × 52C19

Tardigrade Admin · Accepted Answer

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}

Q. 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?

$^{52} C_{18}^{35} C_{2}$

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

$^{52} C_{18} +^{35} C_{2} +^{52} C_{19}$

$^{52} C_{18} \times^{35} C_{2} +^{35} C_{1} \times^{52} C_{19}$

Q. 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?

52C18​35C2​

52C18​×35C2​+52C19​×35C1​+52C20​×35C0​

52C18​+35C2​+52C19​

52C18​×35C2​+35C1​×52C19​

Q. 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?

$^{52} C_{18}^{35} C_{2}$

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

$^{52} C_{18} +^{35} C_{2} +^{52} C_{19}$

$^{52} C_{18} \times^{35} C_{2} +^{35} C_{1} \times^{52} C_{19}$