[Dave H.]

n can be anything and so can r...

The formula n!/(r!*(n-r)!) represents the number of ways you can take n things r at a time where order doesn't matter. For example, if you have a deck of 52 cards and want to know how many two card pockets you could have, you are trying to decide how many ways you can take 52 cards (so n = 52) 2 (so r = 2) at a time. So n = 52 and r = 2 in this example.

The answer is 52!/(2!*(52 - 2)!)

The ! is called "factorial". Don't be scared by the word. It simply means this:

4! = 4*3*2*1
17! = 17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2*1
0! is the only weird one...it is 1 by definition.

So in the above example,
52!/(2!(52-2)!) =
52*51*50*49*...*1/(2*1)*(50*49*48*47*46*...*1)

If you write that out, you see that the 50 in the numerator cancels with the 50 in the denominator. The same goes for 49, 48, 47, etc.

So the only thing you're left with in the numerator that doesn't cancel with something in the denominator is:

52*51

And the denominator still has a 2 in it, but everything else cancelled.

So you're left with 52*51/2 = 26*51 = 1326

Now what if you had 52 cards and wanted to know how many THREE card pockets you could have?

In this case, n = 52 and r = 3
So the formula would be:
52!/(3!*(52-3)!) =
52!/(3!*49!) =

52*51*50*49*...*1/(3*2*1)*(49*48*47*46*...*1)

Notice how 49 in the numerator cancels with 49 in the denominator. The same for 47,46,45,44...all the way down to 1.

So you're left with 52*51*50/3*2*1 = 132600/6 = 22,100

Make more sense now?