How many different ways...

Hey Irene,

Great question - and I love that inquiry about "why" it's true.

You're dead on. There are 12*11*10*9 ways to arrange the sites in order...but we don't care about the order. We only care about the number of groups. And since each group of 4 chosen sites can be arranged in order 4! ways, then we divide by 4! to get from ordered arrangements down to just arrangements.

That gives us the formula:

N!/(K! * (N-K)!) where N = the number of total options and K is the number we select. So you could memorize that, but the above is where that comes from. It's easier to determine the number of ordered groups, so if you divide by the number of ways to order each group, you get to the number of groups.

It's probably also helpful to know that you don't need to do this math! 12*11*10*9/(4*3*2*1) reduces pretty quickly. 4*3 cancels the 12 and 10/2 is 5, so you really have:

11*5*9

So you know that the answer will end in a 5, and only C fits the bill. If you want to check to be sure, 11*45 should be a little over 10*45 (which you know intuitively to be 450), so 495 also works in that way, too. The answer must be C.

12P4 = 11880 ways

Thanks, Brian - great explanation. After sort of getting Maths at school, it really does make a huge difference when you actually understand how it works!