GMAT Math

I've been reviewing permutations, and I'm curious to see if I've mastered some of the basics. How would you evaluate the following?

Let's say you have (and I'm making this example up from scratch):

-- A seven-character license plate
-- The first four characters must each comprise a letter, and no letter can be repeated
-- The last three characters must each comprise a digit, and no number can be repeated

Using the multiplication axiom, would the answer therefore be:

[spoiler]26 * 25 * 24 * 23 * 10 * 9 * 8[/spoiler] ... whatever heinous number that turns out to be? ... which can also, I assume, be expressed as 26P4 times 10P3?

If that's correct ... how would the answer change if we kept all of the same conditions as the above example, but now we decided that the order of the characters doesn't matter, so long as we end up with 4 letters and 3 digits? (And you still can't repeat a character.)

Using the multiplication axiom, would the answer then be:
[spoiler]
36 * 35 * 34 * 33 * 32 * 31 * 30
[/spoiler] ... also expressed as 36P7?

I would approach the first problem with the multiplication principle as you did.

As for the second question, if you use the multiplication principle (ordering matters), you'll have to divide out by repeats. Easier just to use n choose k.

We need 4 letters. There are 26 letters in the alphabeta so there are 26C4 choices.

We need 3 digits. Since there are 10 digits, there are 10C3 choices.

The answer is 26C4 * 10C3.

Your answer for the question allows for a license plate of any amount of numbers or letters.

Cool, thanks for the response and correction. The takeaway here is to recognize when you're dealing with combinations as opposed to permutations, and deal with them accordingly.

Yes, sir. Don't cross the streams.

I'm confused with the 2nd question. Do you mean that order of numbers and letters do no matter, ie, A3C24VF or AAA237A instead of AAAA-123?

Anyways if the question was like that,

I think the answer would be [spoiler]26P4*10P3*7C3
multiply by 7C3 to account for all possibilities for position of letter/number on the plate.
[/spoiler]
Am I right?