Arrangement Of Clothes: Trousers At Ends, Shirt In Middle

Let's dive into a fun combinatorial problem involving arranging clothes! We have five shirts and six trousers, all of different colors. The challenge is to figure out how many ways we can arrange them in a row of eleven display stands, with the special condition that both ends of the row must be occupied by trousers, and the middle position must be occupied by a shirt. This problem combines basic permutation principles with specific constraints, making it an engaging exercise in combinatorial mathematics.

Understanding the Constraints

Before we jump into the calculations, it's crucial to understand the constraints. We have 11 display stands in a row. The first and last positions (the extreme ends) must be occupied by trousers, and the middle position (the 6th position) must be occupied by a shirt. This significantly reduces the number of possible arrangements because these positions are pre-determined in terms of the type of clothing they can hold.

• Ends Occupied by Trousers: This means we need to select two trousers out of the six available to place at the two ends. The order matters because the trousers are of different colors, so placing trouser A at the left end and trouser B at the right end is different from placing trouser B at the left end and trouser A at the right end.
• Middle Occupied by a Shirt: The middle position (6th position) must be occupied by a shirt. We have five shirts to choose from, each of a different color.
• Remaining Positions: After placing trousers at both ends and a shirt in the middle, we have 8 remaining positions to fill. We also have 4 remaining trousers and 4 remaining shirts to arrange in these 8 positions.

These constraints help us break down the problem into manageable steps, making it easier to calculate the total number of possible arrangements.

Breaking Down the Problem

To solve this problem, we'll break it down into several steps:

1. Choosing Trousers for the Ends: We need to select two trousers out of six and arrange them at the two ends. This is a permutation problem because the order matters. The number of ways to do this is denoted as P(6, 2), which stands for the number of permutations of choosing 2 items from a set of 6.
2. Choosing a Shirt for the Middle: We need to select one shirt out of five to place in the middle position. This is a simple selection problem, and the number of ways to do this is 5.
3. Arranging the Remaining Items: After placing trousers at the ends and a shirt in the middle, we have 4 trousers and 4 shirts left to arrange in the remaining 8 positions. This is another permutation problem, but since we have two types of items (trousers and shirts), we need to consider the arrangements of both.

By calculating the number of ways for each of these steps and then multiplying them together, we can find the total number of possible arrangements.

Step-by-Step Calculation

Let's calculate the number of ways for each step:

1. Choosing Trousers for the Ends: The number of ways to choose 2 trousers out of 6 and arrange them is given by the permutation formula:

   $$P(6, 2) = \frac{6!}{(6-2)!} = \frac{6!}{4!} = 6 \times 5 = 30$$

   So, there are 30 ways to place trousers at the two ends.
2. Choosing a Shirt for the Middle: We have 5 shirts, and we need to choose one for the middle position. The number of ways to do this is simply 5.
3. Arranging the Remaining Items: We have 8 positions left, with 4 trousers and 4 shirts to fill them. The number of ways to arrange these 8 items is given by the permutation of the 8 items. This is calculated as 8! (8 factorial).

   $$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$$

   So, there are 40,320 ways to arrange the remaining items.

Combining the Results

To find the total number of possible arrangements, we multiply the number of ways for each step:

$$\text{Total Arrangements} = \text{Ways to arrange ends} \times \text{Ways to place middle} \times \text{Ways to arrange remaining}$$

$$\text{Total Arrangements} = 30 \times 5 \times 40320$$

$$\text{Total Arrangements} = 150 \times 40320$$

$$\text{Total Arrangements} = 6,048,000$$

Therefore, there are 6,048,000 ways to arrange the five shirts and six trousers in a row of eleven display stands such that both the extreme positions are occupied by trousers and the middle position is occupied by a shirt.

Detailed Explanation of Permutations and Combinations

Understanding permutations and combinations is crucial for solving problems like this. Let's delve a bit deeper into these concepts.

Permutations

Permutations deal with the arrangement of items in a specific order. The order of selection matters. The formula for calculating permutations is:

$$P(n, r) = \frac{n!}{(n-r)!}$$

Where:

• n is the total number of items.
• r is the number of items to be arranged.
• ! denotes the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1).

In our problem, when arranging the trousers at the ends, we used permutations because placing trouser A at the left end and trouser B at the right end is different from placing trouser B at the left end and trouser A at the right end. Hence, the order matters.

Combinations

Combinations, on the other hand, deal with the selection of items where the order does not matter. The formula for calculating combinations is:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

Where:

• n is the total number of items.
• r is the number of items to be selected.

The key difference between permutations and combinations is whether the order of selection matters. If the order matters, we use permutations. If the order does not matter, we use combinations.

Why This Problem Uses Permutations

In this specific problem, we primarily use permutations because the arrangement of items matters. For example, the arrangement of trousers at the ends matters, and the arrangement of the remaining shirts and trousers in the middle also matters. If we were simply selecting items without regard to their order, we would use combinations. However, since we are arranging items in a specific order, permutations are the appropriate tool.

Alternative Approaches

While we solved this problem using a step-by-step approach, there are alternative ways to think about it. For example, we could consider all possible arrangements without any restrictions and then subtract the arrangements that violate our constraints. However, this approach would be more complex and prone to errors.

Another approach is to use generating functions, which are a more advanced technique in combinatorics. Generating functions can be used to solve a wide range of combinatorial problems, including those with complex constraints. However, for this particular problem, the step-by-step approach is more straightforward and easier to understand.

Common Mistakes to Avoid

When solving combinatorial problems, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to Consider Order: Always consider whether the order of selection or arrangement matters. If it does, use permutations. If it doesn't, use combinations.
• Double Counting: Be careful not to double count arrangements. This can happen if you're not careful about the constraints.
• Incorrectly Applying Formulas: Make sure you're using the correct formulas for permutations and combinations.
• Ignoring Constraints: Always pay close attention to the constraints of the problem. They can significantly reduce the number of possible arrangements.

By being mindful of these common mistakes, you can improve your accuracy in solving combinatorial problems.

Conclusion

In conclusion, we found that there are 6,048,000 ways to arrange five shirts and six trousers in a row of eleven display stands, given that both ends are occupied by trousers and the middle position is occupied by a shirt. This problem highlights the importance of understanding permutations and combinations, as well as the need to carefully consider the constraints of the problem.

By breaking down the problem into smaller, manageable steps, we were able to systematically calculate the number of possible arrangements. This approach is applicable to a wide range of combinatorial problems and can help you develop your problem-solving skills.

For further reading on combinatorics, check out resources like Khan Academy's Combinatorics lessons. These resources can provide a deeper understanding of the concepts and techniques used in combinatorial mathematics.