Forming 5-Digit Numbers With Constraints A Permutation Problem

Hey there, math enthusiasts! Ever wondered how many different numbers you can create with a specific set of digits, especially when there are certain conditions? Today, we're diving into a fun problem that combines the basics of permutations with a little twist. We're going to figure out how many 5-digit numbers we can make using the digits 3, 4, 5, 6, and 7, but here’s the catch: the digits 4 and 5 have to stick together like glue. Sounds intriguing, right? Let’s break it down step by step and make sure we understand every little detail. This isn't just about crunching numbers; it’s about understanding the logic behind counting possibilities and how these principles can be applied in various scenarios. So, grab your thinking caps, and let's get started!

Understanding the Basics of Permutations

Before we tackle the main problem, let’s quickly revisit what permutations are all about. Permutations are simply different ways we can arrange a set of items. Think of it like lining up your favorite books on a shelf – each different order is a unique permutation. The key thing here is that the order matters. For example, the arrangement '123' is different from '321'. When we're dealing with distinct items (like our digits 3, 4, 5, 6, and 7), the number of permutations of n items taken n at a time is given by n factorial (denoted as n!). Factorial, in mathematical terms, is the product of all positive integers up to that number. So, 5! (5 factorial) is 5 × 4 × 3 × 2 × 1 = 120. This means there are 120 different ways to arrange 5 distinct items.

Now, why is this important for our problem? Well, without any restrictions, if we wanted to find out how many 5-digit numbers we could make with 5 distinct digits, we’d simply calculate 5!. But, as always, math problems love to throw in a little curveball, which in our case is the condition that 4 and 5 must always be together. This seemingly small constraint changes our approach quite a bit. Instead of just arranging individual digits, we need to think of '45' or '54' as a single unit. This is where the fun begins, and we start to see how constraints affect the total number of possible arrangements. This foundational understanding of permutations will help us not only solve this particular problem but also tackle similar counting problems in the future. So, remember, permutations are all about the order, and factorials help us quantify these ordered arrangements.

The Constraint: 4 and 5 Together

Okay, so here’s where things get a little more interesting. Our main challenge is that the digits 4 and 5 need to be together in our 5-digit number. How do we handle this? The trick is to think of the pair '45' (or '54') as a single entity, a sort of dynamic duo that always moves together. By treating '45' as one unit, we reduce our problem from arranging 5 individual digits to arranging 4 entities: 3, 6, 7, and our '45' unit. This simplifies the problem significantly because now we’re dealing with fewer items to arrange. But, we're not quite done yet! Remember, the digits 4 and 5 can be arranged in two ways within their unit: '45' or '54'. This is a crucial detail because each of these arrangements gives us a different set of possible 5-digit numbers.

So, let’s break it down further. First, we consider '45' as a single block. Now we have four entities to arrange: 3, 6, 7, and the '45' block. We can arrange these four entities in 4! ways, which is 4 × 3 × 2 × 1 = 24 ways. But remember, this is only for the '45' arrangement. We also need to consider the '54' arrangement, where 5 comes before 4. The beauty here is that the logic is exactly the same. If we treat '54' as a single block, we still have four entities to arrange: 3, 6, 7, and the '54' block. This also gives us 4! = 24 ways. Therefore, we have 24 arrangements with '45' and another 24 arrangements with '54'. To get the total number of arrangements, we need to add these two possibilities together. Understanding how to treat combined constraints like this is key to solving more complex permutation and combination problems. It's all about breaking the problem down into manageable parts and then putting them back together.

Calculating the Arrangements

Now that we’ve wrapped our heads around the constraint and how to handle it, let’s crunch the numbers and get to the final answer. We’ve already established that we can treat the pair '45' (or '54') as a single unit. This means we are arranging four entities: the digits 3, 6, and 7, along with our '45' or '54' unit. The number of ways to arrange these four entities is 4!, which, as we calculated earlier, is 4 × 3 × 2 × 1 = 24. This is the number of arrangements for each configuration of the pair, either '45' or '54'. But we have two possible configurations for the pair: '45' and '54'. For each of these configurations, we have 24 arrangements. So, to find the total number of 5-digit numbers that meet our criteria, we need to consider both possibilities.

To get the total number of arrangements, we simply add the number of arrangements for '45' and the number of arrangements for '54'. That is, 24 arrangements (for '45') + 24 arrangements (for '54'). This gives us a grand total of 48 different 5-digit numbers that can be formed using the digits 3, 4, 5, 6, and 7, with the condition that 4 and 5 are always together. Isn't it amazing how a simple constraint can change the outcome so dramatically? Without the constraint, we'd have 5! = 120 arrangements, but with it, we're down to 48. This highlights the importance of carefully considering all conditions in a problem. By breaking down the problem into smaller, manageable steps, and by understanding the underlying principles of permutations, we've successfully solved a challenging problem. So, the final answer is 48!

Final Answer: 48

So there you have it, folks! We've journeyed through the world of permutations, tackled a tricky constraint, and emerged victorious with our final answer: 48. It's always satisfying to see how mathematical principles can be applied to solve concrete problems. In this case, we started with a seemingly straightforward question about forming numbers from digits, but we quickly encountered a twist that required us to think creatively. By treating the pair '45' (or '54') as a single unit, we were able to simplify the problem and calculate the number of arrangements. Remember, the key takeaway here is not just the answer itself, but the process we used to get there.

We broke down the problem into smaller parts: understanding permutations, dealing with the constraint, and finally, calculating the arrangements. This approach is super useful not just in math, but in many areas of life. When faced with a complex challenge, try to break it down into smaller, more manageable steps. And don't forget to double-check your work and make sure you've considered all the possibilities! Math, like life, is full of interesting puzzles, and the more we practice, the better we get at solving them. So, keep exploring, keep questioning, and keep having fun with numbers. Who knows what other mathematical adventures await us? Until next time, keep those digits distinct and those pairs together!