2
$\begingroup$

I'm trying to calculate the number of possible password combinations when my password consists of 6 characters that can be

  • uppercase letters
  • lowercase letters
  • digits

I know that if there are no requirements, the amount of possible combinations equals $(26+26+10)^6$, but I am trying to satisfy for the requirement that there must be at least 1 uppercase character, at least 1 lowercase character, and at least 1 digit.

My strategy is to consider that my password can be of the form $(U,L,D,\star,\star,\star)$, with for $U$ and $L$ $26$ possibilities, for $D$ $10$ possibilities, and for $\star$ $62$ possibilities. Then, I only need to correct for their order, but I do not know how to do this.

Any possible steps to take would be very much appreciated.

$\endgroup$
2
$\begingroup$

Use inclusion/exclusio principle:

  • Include the total number of passwords: $(26+26+10)^6$
  • Exclude the number of passwords with no digits: $(26+26)^6$
  • Exclude the number of passwords with no lower-case: $(26+10)^6$
  • Exclude the number of passwords with no upper-case: $(26+10)^6$
  • Include the number of passwords with no lower-case or digits: $(26)^6$
  • Include the number of passwords with no upper-case or digits: $(26)^6$
  • Include the number of passwords with no lower-case or upper-case: $(10)^6$

The answer is therefore:

$$(26+26+10)^6-(26+26)^6-(26+10)^6-(26+10)^6+(26)^6+(26)^6+(10)^6$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.