Question

Determine how many different couples Escort
Dating Service can select if it has 120 men and
210 women registered with the service.

Answer 1

Imagine the men lined up and fixed, from 1 to 120. IF you had a set of 120 women, you have 120 ways to pick the first woman for the first man, 119 ways to pick the second woman for the second man, etc. So if you had 120 men and 120 women, you'd have 120! arrangements of men coupled with women.

Now, from your 210 women, you can pick a set of 120 so you could have what you have above, *but*, you have (210 choose 120) = 210!/[(120!)(210-120)!] possible sets of these 120 women.

So, you have in the end, (210 choose 120) sets of 120 women, each of which can be arranged 120! times with the men, so you have\[\frac{210!}{120!(210-120)!}\times 120!=\frac{210!}{90!}\]different couples.

Answer 2

You can use wolframalpha.com to check out what that number is. It's roughly 7x10^259.