Husband has $12$ relatives $(7 M, 5 F)$ and his wife too has $12$ relatives $(5 M, 7 F)$ . In how many ways can they invite $(6 M, 6 F)$ for a dinner party so that each invites $6$ of his/her relations $7$

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Solution

7M, 5F 5M, 7F
Husband Wife
6M, 0F 0M, 6F
5M, 1F 1M, 5F
4M, 2F 2M, 4F
3M, 3F 3M, 3F
2M, 4F 4M, 2F
1M, 5F 5M, 1F
Total number of ways
$= (^{7} C_{6_{H}} \cdot^{5} C_{0}) \times (^{5} C_{0_{W}} \cdot^{7} C_{6}) + (^{7} C_{5_{H}} \cdot^{5} C_{1})$ $\times (^{5} C_{1_{W}} \cdot^{7} C_{5}) + (^{7} C_{4_{H}} \cdot^{5} C_{2}) \times (^{5} C_{2_{W}} \cdot^{7} C_{4})$
$+ (^{7} C_{3_{H}} \cdot^{5} C_{3}) \times (^{5} C_{3_{W}} \cdot^{7} C_{3}) + (^{7} C_{2_{H}} \cdot^{5} C_{4})$ $\times (^{5} C_{4_{W}} \cdot^{7} C_{2}) + (^{7} C_{1_{H}} \cdot^{5} C_{5}) \times (^{5} C_{5_{H}} \cdot^{7} C_{1})$

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