How many eight character passwords are there if each character is either an uppercase letter A-Z, a lowercase letter a-z, or a digit 0-9, and where at least one character of each of the three types is used?

First how many total 8 character passwords are possible? \(\displaystyle 62^8\)

Second how many 8 character passwords without any digits are possible? \(\displaystyle 52^8\)

Third how many 8 character passwords without any lowercase letters are possible? \(\displaystyle 36^8\)

Finally how many 8 character passwords without any uppercase letters are possible? \(\displaystyle 36^8\)

The answer should be \(\displaystyle 62^8\) - \(\displaystyle 52^8\) - \(\displaystyle 36^8\) - \(\displaystyle 36^8\); correct?

I believe you should apply the principle of inclusion-exclusion in this case; because for example there are some passwords that contain neither numbers nor uppercase letters, and you subtracted these out twice.

So the additional things to count would be

Fifth how many 8 char passwords without any any digits or lowercase? \(\displaystyle 26^8\)

Sixth how many 8 char pass without any digits or uppercase? \(\displaystyle 26^8\)

Seventh how many 8 char pass without any lowercase or uppercase? \(\displaystyle 10^8\)

Final answer:

\(\displaystyle 62^8 -52^8 - 36^8 - 36^8 + 26^8 + 26^8 + 10^8\)