You'll have to forgive me. Math is definitely not my specialty, but my curiosity got the best of me.
My wife watches the show "Are You The One". Doesn't interest me in the least, but it made me wonder about the odds.
The premise, to my understanding, is this:
20 people–10 male and 10 female have to collectively figure out who their 'perfect matches' are.
Each person has one other person who is supposed to be their perfect match.
I was just wondering what the odds are, or how many combinations can be made of groups of 2 from 2 groups of 10.
At first I thought it was 10 factorial (3628800), but then that didn't seem quite right. Question one: Am I correct in assuming this is the number of permutations of 2 groups of 10 divided into 10 groups of 2 with one from each group?
Then, I thought that it was actually 10 * 10 + 9 * 9 + 8 * 8 … + 1 * 1, or 385.
Question 2: Is this the number of combinations of 2 groups of 10 divided into 10 groups of 2, and if so, what is this mathematical operation called (as opposed to factorial)?
Thanks in advance for the help!
Best Answer
This answer is assuming that men can only be paired with women and vice-versa.
Line the men in a line side-by-side and do the same for the women in such a way that the men and women are facing one another and that each couple facing one another is considered a match. Notice that if we leave the men fixed, we can achieve any combination of matchings by simply re-ordering the women of which there are $10!$ ways of doing so in which case your original guess is correct.