Question

A ZIP code contains 5 digits. How many different ZIP codes can be made with the digits 0 through 9 if no digit is used more than once?

Answer 1

that is 10 different numbers, hmm

10 choices for first digit
9 for 2nd (since you used one already)
8 for 3rd
7 for 4th
6 for 5th

so then there are 10*9*8*7*6 different zip codes or 30240 differnt possible zip codes

Answer 2

___ ___ ___ ___ ___

Each dash represents one of the 5 digits of the zip code.

For the first dash there are 10 options for digits. The next dash has 9 options and so on...
Multiply the number of options for each dash:

10*9*8*7*6 = 30240 zip codes

This can also be solved with a Permutation:

nPr
10P5
You could put that into your calculator, but I'll solve it for you.
n!/(n-r)!
10!/(10-5)!
(10*9*8*7*6*5!)/5!
The 5!'s cancel
10*9*8*7*6
=30240 zip codes

It's the same number.