3-Digit Even Numbers: How Many Can You Make?

Hey guys! Today, we're diving into a fun little math puzzle: How many three-digit even numbers can we create using the digits 1, 2, 3, 4, 5, and 6, without repeating any digit? Sounds interesting, right? Let's break it down step by step.

Understanding the Basics

Before we jump into solving the problem, let's make sure we're all on the same page with some basic concepts. A three-digit number has a hundreds place, a tens place, and a units place. An even number is any number that's divisible by 2, meaning it ends in 0, 2, 4, 6, or 8. In our case, we only have the digits 1, 2, 3, 4, 5, and 6 to work with. The phrase "without repeating any digit" means once we use a digit, we can't use it again in the same number. For example, 122 wouldn't be allowed, but 123 would be fine. With that covered, it's time to get to work!

Breaking Down the Problem

Okay, so how do we approach this? The key here is to focus on the units place first. Why? Because the units place determines whether the number is even or odd. Since we want an even number, the units place must be one of the even digits available to us: 2, 4, or 6. So, we have 3 choices for the units place. Once we've chosen a digit for the units place, we move to the hundreds place. We've used one digit already, so we have 5 remaining digits to choose from for the hundreds place. After choosing a digit for the hundreds place, we move to the tens place. We've now used two digits, so we have 4 remaining digits to choose from for the tens place. Therefore, the total number of three-digit even numbers we can form is the product of these choices: 3 (choices for the units place) * 5 (choices for the hundreds place) * 4 (choices for the tens place). Now let's do the math.

Calculating the Possibilities

So, we have 3 choices for the units place (2, 4, or 6). Once we've picked a digit for the units place, we have 5 digits left. Any of these 5 digits can go in the hundreds place. After filling the units and hundreds places, we're left with 4 digits. These 4 digits can be placed in the tens place. To find the total number of possible three-digit even numbers, we multiply these possibilities together: 3 * 5 * 4 = 60. Therefore, there are 60 three-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 without repeating any digit. That's our answer, guys! Now, let's go a little deeper with our example.

Let's Summarize With an Example

Let's say we decide to put '2' in the units place. Now we need to choose a digit for the hundreds place. We could pick '1', '3', '4', '5', or '6'. Let's say we pick '1'. So, our number so far looks like 1_2. Now, for the tens place, we can choose from the digits we haven't used yet: '3', '4', '5', or '6'. If we pick '3', our number is 132. If we pick '4', our number is 142. If we pick '5', our number is 152. If we pick '6', our number is 162. So, just by fixing '2' in the units place and '1' in the hundreds place, we get 4 different three-digit even numbers. We need to do this for every possible combination to find the total number of three-digit even numbers. Remember that permutation and combination are essential in solving these kinds of problems.

Why This Method Works

You might be wondering, why does this method work? Well, it's based on the fundamental counting principle. The fundamental counting principle states that if there are 'm' ways to do one thing and 'n' ways to do another, then there are m * n ways to do both. In our case, we're doing three things: choosing a digit for the units place, choosing a digit for the hundreds place, and choosing a digit for the tens place. By multiplying the number of choices for each of these things, we get the total number of ways to do all three things. Since each of these ways corresponds to a unique three-digit even number, we've found the total number of three-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 without repeating any digit.

Tips for Similar Problems

When tackling problems like this, always start with the most restrictive condition. In our case, the most restrictive condition was that the number had to be even, which limited our choices for the units place. By starting with the most restrictive condition, you reduce the number of possibilities you need to consider and simplify the problem. Also, make sure you understand the problem completely before you start solving it. Read the problem carefully and identify any keywords or phrases that might give you clues about how to solve it. Finally, don't be afraid to break the problem down into smaller parts. By breaking the problem down into smaller parts, you can make it easier to understand and solve. In tackling such problems, practice makes perfect!

Conclusion

So, there you have it! We've successfully calculated that there are 60 three-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, and 6 without repeating any digit. Remember, the key is to break down the problem into smaller, more manageable steps, and always start with the most restrictive condition. Happy problem-solving, everyone! These are the basics to remember when faced with algebra problems!