How can I calculate the number of potential combinations in a password?

Question

If I create a 10 digit password with the following requirements:

• At least one uppercase letter A-Z - 26
• At least one lowercase letter a-z - 26
• At least one digits 0-9 - 10
• At least one common symbol (#,$,%,etc) - 32

By inclusion-exclusion, I can calculate I have ~ 3.2333E+19 possible combinations

However, if I change one of the requirements to at least TWO digits 0-9, how can I calculate the possible combinations?

Answer 1

You have to choose 10 letters, and 2 of them must be digits. Furthermore, there must be one each of a lowercase letter, an upper case letter, and a common symbol. For the others, there are 5 choices to be made, and these are to be made from $26+26+10+32 = 94$ characters.

This gives

$94^5 \approx 7.339e9$

choices for the the 6 other characters that do not have to be digits. And for the digits, there are 2 choices from 10 characters. So this gives

$ 10^2 = 100$

choices. And for the one each of lower-case letters, upper-case letters, and common symbols there are

$ 26 * 26 * 32 = 21632$

choices.

Now lastly, there are $10!$ permutations of these 10 characters, so the total number of combinations of these characters is:

$26*26*32*100*94^5 * 10! \approx 5.76e22$