We have been given the following question for our mock exam:
Each student has a password, which is 6 characters long and each character
is either a digit or a lower case letter. Each password must contain at least
ONE letter. How many possible passwords are there?
26 lower case letters
10 digits
6 characters long
At least one letter
The way I understood, is that it would be easier using inclusion-exclusion, therefore:
$36^6 – 10^6 $ = 2175782336
However another student mentioned it would be:
$36^6 – 10^5 $ = 2176682336
And I also heard:
$36*36*36*36*36*26$ = 1572120576
I think the word "at least ONE letter" is throwing me off a bit. Would anyone mind explaining this to me, please?
Best Answer
Your understanding, and your solution of $36^6 - 10^6$, are correct.
The problem with the answer of $36 \times 36 \times 36 \times 36 \times 36 \times 26$ is that it does not account for the number of different arrangements that are possible. This would be the solution if the password rules were:
However in the actual rules, the restriction on digits is that at least one character has to be a non-digit. There is no specification of which digit this has to be.
Inclusion-exclusion is the perfect way to approach the problem. There are $36^6$ sequences of six digits satisfying the first three rules you posted. The fourth rule,
At least one letter
, rules out all the passwords that are composed entirely of digits. There are $10^6$ of these, so this can be subtracted from the $36^6$ figure. The person who suggested subtracting $10^5$ either made a mistake, or got slightly confused.