We're using the fundamental counting principal to figure out the number of different ways that something can appear. What we're going to look at now is a California license plate and the California license plate one number followed by three letters, followed by three more numbers and what we're asked is to find the find the different number of license plates one with repetition and two without repetition, and basically what that means is with repetition means you're going to have the same letter twice so like a, s where without repetition means once you use the letter a you can't use it again.
So what I do whenever I'm dealing with some problem like where we're filling spots, so I tend to draw a little dashes where I can fill in, so we have our number, letter, letter, number, number, number. And then what I tend to do is think about the number of things that can fill each slot, so this first spot has to be a number. And the common mistake is thinking about how many numbers are, we have the number 0 also, so we have the numbers 1 through 9, but we also have 0 so there's actually 10 numbers that can fill that spot.
We then go to the three letters, there's 26 letters in the alphabet so each of these are going to be 26 followed by three more numbers each number is going to have 10 options so times 10, 10, 10 and the fundamental principal just tells us to multiply all these together. I'm not going to go through that, but what we end up with is basically 10 to the fourth times 26 to the third. It's a big number whatever it is that's fine.
Without repetition basically means we can't repeat the same number over, so again I'm going to draw out my little spots number, letter, letter, letter, number, number, number which should be 7 and then again choosing our first spot we have to choose a number, there's no restrictions because nothing from choosing it, so what we end up with is 10 spots there, 10 things can go.
The next spot is a letter, there's still 26 options, the spot after that is where things starting going as little bit different than previously. We already chose a letter over here it doesn't matter what it is, but that number is available to us any more which means there's still 25 left for that spot so we have 25 by similar logic we've used one letter here, we've used one letter here, so we have two last for that third one leaving us with 24 same concept for the numbers, we used one number here leaving us with 9, then 8 and then 7 and then multiply all these numbers will give us the number of different plates without repetition.
One other thing want to talk about is sometimes these type of problems come up with like phone numbers or something like pretty much the same idea except there is a restriction with that in that often times when you're dealing with your first number on a phone number, you're not going to have that being 0, so just sometimes you're going to get a little restrictions that come up just make sure you think about what your numbers are and where you're going to cause some problems.
Other than that just drawing out your little slots and filling in the numbers that are available in each slot really helps you organize your thoughts.