Help with counting problem

Hi I have struggling to solve what I think is a counting problem. The problem is as follows " A four character code is generated by choosing each character randomly from 26 letters (a,b,c...z) and twenty digits (0,1..20). The password must contain at least one digit and no password can consist of the same character repeated four times. How many passwords are there The part in bold is what I am finding difficult to factor in my combination

HaloWarrior said:

Hi I have struggling to solve what I think is a counting problem. The problem is as follows " A four character code is generated by choosing each character randomly from 26 letters (a,b,c...z) and twenty digits (0,1..20). The password must contain at least one digit and no password can consist of the same character repeated four times. How many passwords are there The part in bold is what I am finding difficult to factor in my combination

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Suppose you put the required digit at the very beginning and ignore the second part (no password can consist ...) for a moment. Can you do that?

If so, suppose you put the required digit in the 2nd, 3rd, or 4th place. The number would be the same as putting it in the 1st place, so the total would be 4 times the first answer [as an aside, the numer to multiply by is from the combination function which in this case is ₄C₁, read as 4 choose 1].

Now, the above also includes all the cases in which you can get a repeated digit four times [since one digit is required in the above, you can't have an alpha repeated four times]. Well how many ways can you do that? Having computed that, subtract the amount from the above.

Denis said:

There are only 10 "digits": 0,1,2,3,4,5,6,7,8,9 ; what do you mean?

If 10 digits and 26 letters (values 1 to 26):
lowest = 0001
highest = zzz9

Agree?

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Hi Denis 10 digits is what I meant
and yes the above combinations are valid

HaloWarrior said:

Hi I have struggling to solve what I think is a counting problem. The problem is as follows " A four character code is generated by choosing each character randomly from 26 letters (a,b,c...z) and ten digits (0,1..9). The password must contain at least one digit and no password can consist of the same character repeated four times. How many passwords are there The part in bold is what I am finding difficult to factor in my combination

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There is really no way to help without doing the problem. But here is my attempt.
1) Explain why there is no need to worry with the at most four times part.HINT, length of code.
2) Explain why there are \(\displaystyle 36^{4}\) ways to make the code with no restrictions.
3) Explain why there are \(\displaystyle 26^{4}\) ways to make the code with no digits.
4) How do we get strings with at least one digit? HINT What is the opposite of none?