In the previous video, we saw a small example with 3 men and 3 women. We defined a “match” as an assignment of each man to exactly one woman, and each woman to exactly one man. The optimal match in the previous video was to assign man 1 to woman 3, man 2 to woman 1, and man 3 to woman 2.
How many different feasible matches are there in the example with 3 men and 3 woman? (HINT: Another feasible match is to assign man 1 to woman 1, man 2 to woman 3, and man 3 to woman 2.)
How many different feasible matches are there with 5 men and 5 women? (HINT: First assign man 1 to one of the women. How many choices are there? Then assign man 2 to a woman - how many choices are there now? Repeat this until every man is matches to every woman.)
In the first case, there are 6 possible matches. The first man can be assigned to any of the 3 women (3 choices). Then the second man can be assigned to any of the remaining 2 women (2 choices). The third man is automatically assigned to the remaining woman. So there are a total of 3*2 = 6 choices.
In the second case, there are 120 possible matches. The first man can be assigned to any of the 5 women (5 choices), then the second man can be assigned to any of the remaining women (4 choices), etc. This gives a total of 5*4*3*2 = 120 different matches.
You can easily see how the number of possible matches gets very large on online dating sites!