You will be invited, Maybe
Ashish has 7 relatives, 4 of them are ladies and 3 gentlemen, his wife 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 Ashish's relative and 3 of his wife's relative?
The answer is 485.
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1 solution
Husband's male relatives (HM) : 3
Husband's female relatives (HF) : 4
Wife's male relatives (WM) : 4
Wife's female relatives (WF) : 3
The conditions are :
HM + WM = 3
HF + WF = 3
HM + HF = 3
WM + WF = 3
By the rules, they can invite these relatives in 4 major different ways :
1) HM : HF : WM : WF = 3 : 0 : 0 : 3
2) HM : HF : WM : WF = 2 : 1 : 1 : 2
3) HM : HF : WM : WF = 1 : 2 : 2 : 1
4) HM : HF : WM : WF = 0 : 3 : 3 : 0
Answer
= (3C3)(4C0)(4C0)(3C3) + (3C2)(4C1)(4C1)(3C2) + (3C1)(4C2)(4C2)(3C1) + (3C0)(4C3)(4C3)(3C0)
= 1 x 1 x 1 x 1 + 3 x 4 x 4 x 3 + 3 x 6 x 6 x 3 + 1 x 4 x 4 x 1
= 1 + 144 + 324 + 16
= 485