In how many ways can 5 men, 4 women, and 3 children be arranged in a row of 12 seats is the men sit together?

To solve the problem of finding the number of ways to arrange 5 men, 4 women, and 3 children in a row of 12 seats such that the 5 men sit together, we can break down the problem as follows:

### Step 1: Treat the group of men as a single unit.
Since the 5 men must sit together, we can treat them as a single "block." This reduces the problem to arranging 8 units: the "block" of men, 4 women, and 3 children.

### Step 2: Arrange the units.
The 8 units can be arranged in \(8!\) (8 factorial) ways. So, the number of ways to arrange these units is:
\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\]

### Step 3: Arrange the men within their block.
Within the "block" of men, the 5 men can be arranged among themselves in \(5!\) (5 factorial) ways. So, the number of ways to arrange the men within their block is:
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Step 4: Multiply the two results.
The total number of arrangements is found by multiplying the number of ways to arrange the 8 units by the number of ways to arrange the men within their block:
\[
8! \times 5! = 40320 \times 120 = 4,838,400
\]

So, the total number of ways to arrange the 5 men, 4 women, and 3 children in a row of 12 seats such that the men sit together is **4,838,400**.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What if the women must also sit together in addition to the men?
2. How many ways can the men be arranged if one specific man must always sit in the middle of the block?
3. How would the solution change if there were an additional restriction that two specific children must not sit next to each other?
4. How many ways can the 4 women and 3 children be arranged if there are no restrictions on the men's seating?
5. What if the men, women, and children must alternate seats in the arrangement?

**Tip:** When solving combinatorial problems involving restrictions, consider using the strategy of treating groups as single units to simplify the counting process.

Explore how to arrange 5 men, 4 women, and 3 children in a row of 12 seats where the men must sit together. This combinatorial problem involves using factorial notation and permutation concepts. Learn step-by-step calculations to determine the total number of arrangements.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation