License Plates Starting With Z And Ending In 7: How Many?

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How Many License Plates Can Start with Z and End in 7?

Hey guys! Ever wondered just how many license plates we can actually create if they have to follow a specific pattern? Let's dive into a fun little math problem where we figure out exactly that! We're going to look at license plates that start with the letter 'Z' and end with the number '7'. Sounds specific, right? Well, let's break it down and see how many possibilities we've got.

Understanding the License Plate Structure

First things first, we need to understand what a typical license plate looks like in this scenario. Our license plates have a pretty standard format: three letters followed by two digits. Think of it like this: [Letter][Letter][Letter][Digit][Digit]. We also know a key piece of information: our alphabet has 27 letters. This is super important because it tells us the range of letters we can use.

So, why is this format so important? Well, each position in the license plate has a certain number of options. By figuring out how many options each position has and then multiplying them together, we can find the total number of possible license plates. This is a classic example of using the fundamental counting principle, which is a fancy way of saying we multiply the possibilities for each step to get the total possibilities. It's like building a combination lock; each dial has a certain number of settings, and the total number of combinations is the product of those settings.

Breaking Down Each Position

Let's take it position by position. The first three slots are for letters, and since we're using a 27-letter alphabet, each of these slots could theoretically have any of those 27 letters. The last two slots are for digits, which typically range from 0 to 9, giving us 10 possibilities for each digit slot. However, we have a twist in our problem: we're not looking at just any license plates. We've got specific requirements!

The 'Z' and '7' Constraints

Okay, now for the fun part! We've got some rules to follow. Our license plates must start with the letter 'Z' and must end with the digit '7'. This is where things get a little more interesting. These constraints dramatically reduce the number of possible license plates. Instead of every slot being a free-for-all, some slots are now fixed.

Impact on the First and Last Positions

The fact that the first letter must be 'Z' means we don't have 27 options for that first slot anymore. We only have one option: 'Z'. Similarly, the last digit must be '7', so we only have one option for that last digit slot as well. These fixed positions act like anchors, significantly simplifying our calculation. We've essentially reduced the problem to figuring out the possibilities for the remaining slots, which makes our job way easier!

Calculating the Possibilities

Alright, let's get down to the nitty-gritty and calculate how many license plates we can actually make. Remember our license plate format: [Z][Letter][Letter][Digit][7]. We know the first position is fixed as 'Z' and the last position is fixed as '7'. That leaves us with three positions to figure out: two letter positions and one digit position.

Second and Third Letter Positions

For the second and third letter positions, we can use any of the 27 letters in our alphabet. There are no restrictions here, so we have full freedom! This means we have 27 options for the second letter and 27 options for the third letter. Think of it as having a full deck of cards to choose from for each of these positions.

The Fourth Digit Position

The fourth position is a digit, and again, there are no restrictions. We can use any digit from 0 to 9, which gives us 10 options for this position. It's like having a 10-sided die and any roll is valid for our license plate.

The Final Calculation: Multiplying It All Together

Now for the magic step! To find the total number of possible license plates, we multiply the number of options for each position together. We have:

  • 1 option for the first letter (Z)
  • 27 options for the second letter
  • 27 options for the third letter
  • 10 options for the first digit
  • 1 option for the second digit (7)

So, the total number of license plates is 1 * 27 * 27 * 10 * 1 = 7290. Ta-da! We've got our answer.

Conclusion: The Grand Total

So, there you have it! We can register a maximum of 7,290 license plates that start with the letter 'Z' and end with the number '7', given our three-letter, two-digit format. Isn't it amazing how a few simple constraints can lead to a specific, calculable number? This kind of problem shows how math can be used in everyday situations, even in something as mundane as license plates. Hopefully, you guys found this breakdown interesting and maybe even a little fun! Keep those questions coming, and we'll keep cracking the code!

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