[Math] Unique Digit Password Probability Problem

combinatoricspermutationsprobability

Kevin has to make a 7-character password. For each scenario, calculate the total possible outcomes.

  1. Each character must be a digit 0-9.
  2. 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.

Best Answer

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.

  1. 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 $10 \times 7$).

  1. 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: $26 \times 25 \times 24 \times 10^4$.

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