Hi folks, two quick combinatorics questions here I can't figure out:

1.Assume that a car licence plate can consist of up to 6 characters. Each character can be any of the letters from A to Z, or any numeral from 0 to 9. How many licence plates are possible?

2. Write this expression without using the factorial symbol: (n-3)! / n!

Question

Hi folks, two quick combinatorics questions here I can't figure out:

1.Assume that a car licence plate can consist of up to 6 characters. Each character can be any of the letters from A to Z, or any numeral from 0 to 9. How many licence plates are possible?

2. Write this expression without using the factorial symbol: (n-3)! / n!

anonymous · Answer

Question 1 is simply 36^6. 36 possible characters, 6 spots for them.
Question 2 can be written as 1/n(n-1)(n-2), the other terms all cancel out.

nikolas · Answer

Thanks nbouscal. I think I've figured out the second one, but still a little unclear about the first. The question asks for the number of plates with *up to* 6 characters, not exactly 6.
 I tried: $$36^{6}\times36^{5}\times36^{4}\times36^{3}\times36^{2}\times36^{1}$$ which is 4.8123e32, but the answer key says "223897 * 6116" which I assume is either the product, 1369354052, or a typo...

anonymous · Answer

Change those multiplications to additions and I think you may be right.

nikolas · Answer

Ah, that must be it. Adding them gives 2238976116 so I'll assume the "X" in the answer key is a typo. Thanks!