Unique Arrangement Problem: When All Men Stand Next to Each Other

Imagine a scenario where you have a line-up of individuals consisting of 8 women and 5 men. The challenge is to find the number of unique arrangements where all 5 men are standing together. This problem involves a combination of permutations and factorials, providing an engaging insight into combinatorial mathematics.

Understanding the Problem

The problem can be broken down into smaller, manageable steps. Here’s a detailed explanation:

Step 1: Treating the Men as a Single Unit

In this step, we treat the entire group of 5 men as a single unit or block. This simplifies the problem by reducing the number of units to be arranged. With this block, we now have a total of 9 units to arrange: 8 individual women plus the single block of 5 men.

Step 2: Calculating the Arrangements of Units

We need to find out how many ways we can arrange these 9 units. The mathematical way to do this is by calculating the factorial of 9, denoted as 9!. Factorial is the product of all positive integers up to the given number. For 9!, we have:

9! 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 362,880

Step 3: Arranging the Men Within Their Block

Inside the block of 5 men, there are 5 different men who can be arranged among themselves. The number of ways to do this is given by the factorial of 5, denoted as 5!. For 5!, we have:

5! 5 × 4 × 3 × 2 × 1 120

Step 4: Calculating the Total Arrangements

To find the total number of arrangements where all the men are standing next to one another, we multiply the number of ways to arrange the units (9!) by the number of ways to arrange the men within their block (5!). Thus, the calculation is:

9! × 5! 362,880 × 120 43,545,600

Conclusion

Therefore, the total number of unique arrangements where all 5 men are standing next to each other, in a line of 8 women and 5 men, is 43,545,600.

Simplified Explanation

The simplified way to calculate this is to treat the 5 men as one unit, which leaves us with 9 units (8 women 1 block of 5 men). The number of ways to arrange these 9 units is 9!. Then, the number of ways to arrange the 5 men within their block is 5!. Multiplying these two values gives us the total number of arrangements.

Example Calculation

A different approach to view this problem is to consider that the block of 5 men can be placed in any of the 9 positions. There are 5! ways to arrange the men within the block and 8! ways to arrange the 8 women. Therefore, the answer can also be calculated as:

(9 × 5!) × 8! 43,545,600

Alternatively, another method is to create one man in 5! ways, then there are 9 “people” to arrange in 9! ways. The total number of arrangements is calculated as:

5! × 9! 43,545,600

Conclusion Recap

In summary, the problem of arranging a line of 8 women and 5 men such that all men are standing together can be solved by treating the 5 men as a single unit, then calculating the permutations and factorials involved. The final answer, using factorials and permutations, is 43,545,600 unique arrangements.