Question

A man has 7 relatives in which 4 are ladies and 3 are gentlemen. His wife also has 7 relatives in which 3 are ladies and 4 are gentlemen. In how many ways can the couple invite the relative for a dinner party, 3 ladies and 3 gentlemen, so that 3 of them are man’ relatives and 3 of them are his wife’ relatives?

Answer 1

The possible number of ways can be listed as follows.

| Man | wife | Number of ways |
| Ladies (4) | Gentlemen (3) | Ladies (3) | Gentlemen (4) | |
| 3 | 0 | 0 | 3 | 4C3 . 4C3 = 16 |
| 2 | 1 | 1 | 2 | (4C2 . 3C1)2 = 324 |
| 1 | 2 | 2 | 1 | 4C1 . 3C2)2 = 144 |
| 0 | 3 | 3 | 0 | 3C3 . 3C3 = 1 |
| | | | Total | 485 |

Therefore, total number of ways of inviting relatives for a dinner party of six = 485.