Zermelo
Junior Member
- Joined
- Jan 7, 2021
- Messages
- 148
Hello,
I'm working on a complicated combinatorics problem from my Discrete Math class. The problems goes like this:
The husband has 12 cousins: 5 men and 7 women
His wife also has 12 cousins: 7 men and 5 women.
They don't have any common cousins.
They want to invite 6 husband's cousins and 6 wife's cousins, but such that there will be 6 men and 6 women at the party (excluding the husband and wife).
How many ways are there to do that?
Well, I drew myself a sketch and managed to "brute force it", what I got is:
[MATH]{7\choose 6}{7\choose 6} + {7\choose 5}{5\choose 1}{5\choose 1}{7\choose 5} + {7\choose 4}{5\choose 2}{5\choose 2}{7\choose 4} + {7\choose 3}{5\choose 3}{5\choose 3}{7\choose 3}+ {7\choose 2}{5\choose 4}{5\choose 4}{7\choose 2} + {7\choose 1}{5\choose 5}{5\choose 5}{7\choose 1}[/MATH]
My logic was: the first solution to invite all the women from the husband's family and all the men from the wife's, the second solution is to invite 6 women from the husband's and 1 from the wife's and that mean's 1 man from the husband's and 5 from the wife's etc etc.
My first question is: is my line of reasoning ok? Is this a correct use of the product rule?
The second question is: is there a simpler way of doing this? Maybe I can translate this problem to combinations or permutations with repetition? Or is this the only way?
Thanks in advance!
I'm working on a complicated combinatorics problem from my Discrete Math class. The problems goes like this:
The husband has 12 cousins: 5 men and 7 women
His wife also has 12 cousins: 7 men and 5 women.
They don't have any common cousins.
They want to invite 6 husband's cousins and 6 wife's cousins, but such that there will be 6 men and 6 women at the party (excluding the husband and wife).
How many ways are there to do that?
Well, I drew myself a sketch and managed to "brute force it", what I got is:
[MATH]{7\choose 6}{7\choose 6} + {7\choose 5}{5\choose 1}{5\choose 1}{7\choose 5} + {7\choose 4}{5\choose 2}{5\choose 2}{7\choose 4} + {7\choose 3}{5\choose 3}{5\choose 3}{7\choose 3}+ {7\choose 2}{5\choose 4}{5\choose 4}{7\choose 2} + {7\choose 1}{5\choose 5}{5\choose 5}{7\choose 1}[/MATH]
My logic was: the first solution to invite all the women from the husband's family and all the men from the wife's, the second solution is to invite 6 women from the husband's and 1 from the wife's and that mean's 1 man from the husband's and 5 from the wife's etc etc.
My first question is: is my line of reasoning ok? Is this a correct use of the product rule?
The second question is: is there a simpler way of doing this? Maybe I can translate this problem to combinations or permutations with repetition? Or is this the only way?
Thanks in advance!