Question

7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife also has 7 relatives; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relative and 3 of the wife’ s relatives?

Answer 1

Number of relatives of a man = 7

4 ladies and 3 gentlemen

Number of relatives of the man’s wife = 7

3 ladies and 4 gentlemen

The dinner party consists of 3 ladies and 3 gentlemen.

For the dinner party,

3 persons from the man’s relatives and 3 persons from the man’s wife’s relatives are invited.

Then we have the following possibilities for the different possible arrangements.

	M	W	M	W	M	W	M	W
Ladies	0	3	1	2	2	1	3	0
Gentlemen	3	0	2	1	1	2	0	3

Required number of ways

= (⁴C₀)(³C₃) x (³C₃)(⁴C₀) + (⁴C₁)(³C₂)(³C₂)(⁴C1) + (⁴C₂)(³C₁)(³C1)(⁴C₂) + (⁴C₃)(³C₀)(³C₀)(⁴C₃)

= `1 xx 1 xx 1 xx 1 + 4 xx 3 xx 3 xx 4 + (4!)/(2!(4 - 2)!) xx (3!)/(1!(3 - 1)!) xx (3!)/(1!(3 - 1)!) xx (4!)/(2!(4 - 2)!) + ""^4"C"_1 xx 1 xx 1 xx ""^4"C"_1`

= `1 + 144 + (4 xx 3 xx 2!)/(2! xx 2!) xx (3 xx 2)/(2!) xx (3 xx 2!)/(2!) xx (4 xx 3 xx 2!)/(2! xx 2!) + 16`

= `145 + (4 xx 3)/(2 xx 1) xx 3 xx 3 xx (4 xx 3)/(2 xx 1) + 16`

= 145 + 2 × 3 × 3 × 3 × 2 × 3 + 16

= 145 + 324 + 16

= 485