# Math Help - Eight character passwords

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? $62^8$

Second how many 8 character passwords without any digits are possible? $52^8$

Third how many 8 character passwords without any lowercase letters are possible? $36^8$

Finally how many 8 character passwords without any uppercase letters are possible? $36^8$

The answer should be $62^8$ - $52^8$ - $36^8$ - $36^8$; correct?

2. Originally Posted by oldguynewstudent
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? $62^8$

Second how many 8 character passwords without any digits are possible? $52^8$

Third how many 8 character passwords without any lowercase letters are possible? $36^8$

Finally how many 8 character passwords without any uppercase letters are possible? $36^8$

The answer should be $62^8$ - $52^8$ - $36^8$ - $36^8$; 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? $26^8$

Sixth how many 8 char pass without any digits or uppercase? $26^8$

Seventh how many 8 char pass without any lowercase or uppercase? $10^8$

$62^8 -52^8 - 36^8 - 36^8 + 26^8 + 26^8 + 10^8$

3. Originally Posted by undefined
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? $26^8$

Sixth how many 8 char pass without any digits or uppercase? $26^8$

Seventh how many 8 char pass without any lowercase or uppercase? $10^8$

$62^8 -52^8 - 36^8 - 36^8 + 26^8 + 26^8 + 10^8$