I have a question regarding the following problem.
A crime detective is able to dust a lock for finger prints and he determines that the numbers: 3,4,5 and 8 have been pressed repeatedly and the other numbers have not. If the lock is a standard 10 digit number pad, and the combination contains 5 places, how many different possible password could he have to try before finding the correct one?
The following is my claim.
It is natural to think that the four numbers are pressed repeatedly, therefore each numbers are used at least once.
So, using the fundamental counting principle with 5 places, I would say the first number could be any of the 4 numbers.
This is where I get iffy.
The next number may or may not be the repetition of the previous, so I am not sure whether to call it 4 choices or 3 choices.
Ignoring my confidence, I think the answer is $4*4!$ which implies the fact that there are 24 combinations of numbers with 4 choices of numbers that could have been repeated.
What concerns me is the position of the repeated number, though.
I know that the combinations of the lock is sensitive to order, so I am not sure if the second number is repeated, third number is repeated, etc.... and I cannot tell numerically if that makes a difference or not.
Can I have some confirmation?