Okay for this problem we're going to be looking at on all the different ways you can make a license plate is a very tip ability problem. So we're gonna look at first is that we've got right now what we know here. We know that, um, for automobile license plates, we can have different letters followed by different numbers. What I do with my students is I get one example some here's an example. This kind of gets depart, see what they say. Well, letter A. We talk about letters being done, so maybe you'll have X y Z and your license plate, and then maybe we'll have three different number. Possibility is X Y Z and 123 That's an example, right? So part A says, How many different ways are there to have ah, three letter combination? It's really three letter arrangement because the order is the order matters in this case. So X y Z is different than xz. Why? So that affects the math a little bit. So we do all the different possibilities. So let's look apart. A So, um and it does say the state does not use valve or the letter y so Oh, shoot. I mean, stick there. So I'm just gonna just for reference you We couldn't have X y Z because they say no vowels and no letter. Why? So let's see wells and know why is the reason being that out? Because that affects our number possibility. There's 26 letters in the alphabet, so they're gonna basically knock out the five vowels in the letter y So we could say I can't use that. So maybe we'll be X daisy. That's what that means there. So what that means is that we have 20 letters combinations, 20 to pick from. So for part, A says how many different each letter can only appear once on any given license plate. How many different sets of three letters or possible? So if you only do it once Well, for your first pick, you have 20 different pics. And for your second pick, you can't use that. Whatever you pick again, let's say you pick an extra can't pick use X again. That drops you down to 19 picks. And for the third combination, that's 18 letters to choose from, so people play all three of those together that means that you have 6840 different arrangements. So that's party. Let's look apart being they say, Well, how many different ways of the new numbers. So, for part B, we know that we have 10 digits were picking three. There were zero accounts of a digit. All right, so for digits have been different. Three letter digits. Can we have three letter combinations I should needs were combination that usual letter word arrangements cause border matters and not repeating matters. That's why the numbers go down. So that's three number speed digits. So my example there, 123 could be some 1012 could be another one. But in general, we know that we have 10 different digits that we could pick. And then ah, I called the domino effect. So you have 10 you pick a certain number and then you're a number of different numbers to pick from was down by one and the 3rd 1 goes down by another one. So you do 10 times nine times eight. So if you 10 times nine times eight, that gives you 720 different arrangements of three digits, so That's the answer for part B. Important sees where you put it all together. So part C. Wherever sand. Okay, we're gonna put all the letters together, so we think of all the different license plates. And the whole purpose of that is we have different license plates with three letters and three numbers. So we're looking for the number of about license plates. Their possible unique license plates, like your parents have a license plate that's different than the neighbors license plates were really doing here is taking these together. So really, them? I'm not going to read you the math, but we've done it for parts A and B. So you take 2010 19 times 18 times, 10 times, nine times eight. Because we've already partially done the numbers, we just have to multiply those two together. So what we have to do This just takes 6840 combinations of the letters Time. 720 digit combinations. Overall, our final final answer. Millions of your state has all these possibilities for everybody to have a unique license plates. So four million and 24,000 800 different license plates. That's a nice story, a little problem. So that's how they do it. All right, so you get the arrangements. Big thing. Where there was watched what I call a domino effect. It goes down each time. He's not repeating letters, and we had 26 letters in the alphabet, but no vowels in a way.