Unique Digit Password Probability Problem Kevin has to make a 7-character password.  For each scenario, calculate the total possible outcomes.


*

*Each character must be a digit 0-9.

*The first three characters must be upper-case letters, but each must be unique. The remaining four characters must be digits 0-9.


I'm not completely sure if I'm setting these up the right way so please let me know.
1. This one I simply did 7 characters * 10 digits = 70 possible outcomes.
2. For this one I know there's 26 letters in the alphabet but the first three need to be unique so that would be 26-3=23. That is 23 over 3 positions which is (23 3) possible outcomes + (4*10) since the remaining 4 characters are multiplied by 10.
 A: HINT: The possible outcomes for each digit must be multiplied, not added.
A: for the first, you have to choose a number between $0000000 $ and $9999999$ which gives $10^7$ possibilities.
for the second, to choose $3$ different between $26$ and a number between
$0000$ and $9999$.
this allows 
$6\binom{26}{3}.10^{4}$
$=26.25.24.10^{4}$  choices.
A: You kind of have the right idea but I think some conceptual issues. 
You want a 7 character password under the following constraints and want to know how many total possibilities there are. With problems like this, it is useful to think of each character as a slot and for each slot, if you calculate the number of choices you have and multiply all the slots together, you will get the number of total possibilities.


*

*Each character must be a digit 0-9.


You have $7$ characters and each one can be a digit from 0-9. So there are $10$ possible choices for the first character (slot), $10$ possible choices for the second character (slot) etc. So you have $10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^7$ possibilities (not $10 \times 7$).


*The first three characters must be upper-case letters, but each must be unique. The remaining four characters must be digits 0-9.


We do the same thing. How many possibilities are there for the first character (slot)? Well we have $26$ uppercase letters so it is $26$. What about the second character (slot)? We have already used one of our $26$ letters so we only have $25$ possible letters to pick from so for the second character (slot), we have $25$ choices. Similarly for the third character, we have used up $2$ of the $26$ letters in the alphabet so we have $24$ possible choices for the third character (slot). The remaining characters can be digits from 0-9 so we have $10$ possibilities for the fourth, fifth, sixth, and seventh character (slot).
So total number of ways is: $26 \times 25 \times 24 \times 10^4$.
