Arranging 7 Women and 3 Men in a Row with the Condition that Men Must Stand Next to Each Other

When arranging a group of 7 women and 3 men in a row with the condition that the men must stand next to each other, the problem can be simplified by treating the group of 3 men as a single unit. This article explores the solution step-by-step and provides insight into the mathematical principles behind the arrangement.

Step-by-Step Solution: Treating Men as a Single Unit

The problem can be broken down into four steps, each focusing on a different aspect of the arrangement:

Step 1: Treat the Men as a Single Block

Since the 3 men must stand together, we can consider them as a single unit. This unit consists of 3 men, and when we factor in the 7 women, we are essentially arranging 8 blocks. Therefore, the number of ways to arrange these 8 blocks is given by 8 factorial (8!):

8! 40320

Step 2: Arrange the Men Within Their Block

Within this block of 3 men, they can be arranged in different ways. The number of arrangements for 3 men is given by their factorial (3!):

3! 6

Step 3: Calculate the Total Arrangements

To find the total number of arrangements, we multiply the number of ways to arrange the 8 blocks by the number of ways to arrange the 3 men within their block:

Total arrangements 8! * 3! 40320 * 6 241920

Alternative Method: 8 Slots and Choices

Another way to approach the problem is to consider the 3 men as a single unit and treat the entire group as an 8-element set, where each element can be either one of the 7 women or the group of 3 men.

Step 1: Fill the First Slot

The first slot can be filled in 8 different ways: either one of the 7 women or the group of 3 men.

Step 2: Fill the Second Slot

After filling the first slot, 7 different options remain. Therefore, the second slot can be filled in 7 ways.

Combining the first and second slots, there are 8 * 7 56 ways to fill the first two slots.

Continuing in this manner, the total number of ways to fill all 8 slots is:

8! 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 40320

Step 3: Arrange the Men Within Their Block

The group of 3 men can be ordered in 3! different ways:

3! 3 * 2 * 1 6

Combining the number of ways to arrange the 8 slots and the number of ways to arrange the 3 men within their block, we get the total arrangements:

8! * 3! 40320 * 6 241920

Conclusion

In summary, the total number of ways to arrange 7 women and 3 men in a row with the condition that the men must stand next to each other is 241,920. This solution demonstrates the power of factorial notation and permutation calculations in solving complex arrangement problems.

Keywords

arrangement, permutation, factorial, men standing together