3 9 points suppose license plates in a certain state consist...

3. (9 points) Suppose license plates in a certain state consist of three letters followed by three digits. (Letters are chosen from $\{A, B, \ldots, Z\}$ and digits are chosen from $\{0, 1, 2, \ldots, 9\}$.) You can leave your final answers as mathematical expressions, such as $\binom{5}{3}$, $8!/10!$, or $5^3$. No need to evaluate them numerically. (a) (3 points) How many different license plates can be made? (b) (3 points) What is the probability that a randomly chosen license plate contains at least two distinct letters (i.e., the same letter does not appear three times in the chosen plate)? (c) (3 points) A new law was just put into place banning both the letter 'O' and the digit '0' to prevent ambiguity when reading the plates. How many different license plates can be made under this new law?

Transcript

00:01 So in this question, we're asked to deal with the intricacies of license plate law, and what we need to know here is just our multiplication counting rule, which states that if there is a way of doing one thing, and then we have b ways of doing another thing, sorry, of doing another thing, then there are a times b ways of performing both of these things, okay? so if there are five ways for you to pick a fruit, and five ways for you to pick a cookie, there are five times five ways or 25 ways for you to pick both, a fruit and a cookie.

00:31 Okay? so let's apply that here.

00:35 In part a, we're told that in a certain state, license plates consist of three letters, a to z, followed by three digits, zero to nine.

00:42 What we're asked to do here is to determine how many different license plates can be issued.

00:46 Okay? so i know that where i'm from, we're actually allowed to repeat our digits and numbers and our license plates.

00:53 So this means that we're going to be able to use our multiplication counting rule because our events aren't really connected, you know? just because you pick zero for your first digit doesn't mean it's going to affect what happens when you pick your next digit, right? so if you were to break down our license plate, we see that we have six spots, right? these three over here are reserved for letters, and these three right here are reserved for letters.

01:31 For numbers or digits.

01:36 Okay? we know that there are 26 letters in the alphabet, and over here we are given 10 digits, okay? so, using our multiplication counting rule, we know that if there's a certain number of ways to choose our letters, and there's a certain number of ways to choose our digits, then the number of ways we can choose both our letters and our digits is going to be the number of letters, sorry, the number of ways to choose letters, multiplied by the number of ways to choose digits, right? but if we take that a step further and apply that to figuring out how we can pick our letters and our digits, what we're going to see is that, so if we do that, we know that there are 26 ways to choose the first, 26 ways to choose a second, and 26 ways to choose a third, right? so it means that we'll have 26 times 26 times 26.

02:39 Okay? and for our 10 digits, there are going to be 10 digits to choose from every time.

02:44 So we're going to have 10 times 10 times 10.

02:49 This is going to give us the total number of combinations for our license plates.

02:57 So let me just, let's plug this into the calculator.

03:01 We're probably going to get an answer in scientific notation because it's going to be a pretty big answer, i think.

03:10 Oh, or not.

03:11 Okay.

03:11 Okay, so it's in the millions.

03:13 We have 17 ,000, 576 ,000.

03:21 Okay? so it wasn't as big as i was predicting.

03:29 But yeah, that's our part a.

03:37 So if you move on to part b, we are asked for, sorry, let me just move this over.

03:52 In part b, we are asked that if our state allows any six character mix in any order of 26 letters in 10 digits, how many unique plates are possible? so now we're not going to be dealing with a situation like we did in part a, right? where we divided everything into letters and digits.

04:13 This time, what we're going to be doing is that, like, you know, we're going to have our six slots, and this time we aren't going to be dividing them based on letters and numbers.

04:28 Now, we're allowed to use whatever we want.

04:31 So it means that our total is not 26 or 10, but rather it's 26 plus 10, because that's the number of letters and digits we have available to choose from.

04:41 So that's going to be 36.

04:42 Now, we know that for each of these slots up here, there are 36 ways to choose.

04:49 And since we know that we are allowed to repeat, if you don't in your area, well, i'm pretty sure they do, actually, in most areas, they allow you to repeat.

05:03 So i wouldn't worry about anything else, okay? so we have 36 times 36 times 36 times 36 times 36 and i'm writing a bit bigger than i gave myself space for times 36.

05:24 But what it's basically going to be is 36 to the power of 6.

05:29 And so if we do that, we see that we are going to get 2 billion, 176 million, 786 ,000, 7802 ,000...

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