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in Permutations and combinations by (52.7k points)

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’s relatives and 3 of them are his wife’s relatives?

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Best answer

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.

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