Question

You are asked to select a password for a Web site. It must consist of five lowercase letters and two digits in any order. How many possible such passwords are there if a) repetitions are allowed b) repetitions are not allowed?

Answer 1

Since there must be 5 lower case characters and 2 digits so the password is 7 characters long.

(a) repetitions are allowed. Therefore there are 26^5 lowercase characters and 10^2 digits that we can pick.

Now because The digits can be placed in any order we may pick the positions of the lowercase characters or the digits. The 5 lowercase characters may be placed in 5 of the 7 slots 7C5=21 ways.

We don't need to do the calculation for the digits into the two remaining slots because they will automatically fall into the two remaining places.

So there are 21 * 26^5 * 10^2 = 2.5 * 10^10 passwords possible.

(b) repetitions are not allowed. Therefore there are 26P5 selection of lowercase characters, and 10P2 possible digits possible. The way we pick our possible positions are still the same 7C5=21ways .

So there are 21 * 26P5 * 10P2 = 1.5 * 10^10 passwords possible.