If I have 6 red, 6 blue and 6 green blocks, how many ways can I arrange all 18?

If you can distinguish one from another, say by numbers on them, then you would have 18 ways to choose the first one, 17 for the next, etc. 18x17x16x...x1 = 18! = a very big number.

But presumably you cannot distinguish between blocks of the same color. So you have to account for the "duplicates" in the 18!. The 6 red blocks can be arranged (in whatever places they end up), in 6x5x4x3x2x1 = 6! = 720 ways. So you divide by 6!

You do the same for the blue and green, and you end up with 18! divided by (6! x 6! x 6!).

That's equal to 6,402,373,705,728,000 / 373248000

= 17,153,136

The formula is very easy, and applies to questions like "how many words can be formed by rearranging the letters in "MISSISSIPPI"?

You just count the letters, and look for sets of repeats.

MIS SIS SIP PI has 11 letters, 4 S's 3 I's and 2 P'sl. So you take 11! and divide by 4! x 3! x 2!.

In general, when you have n letters, or colored blocks, or whatever, you take n! and divide by k! for every distinct set of k repeated things.

18! / (6! 6! 6!) = 17,153,136 ways