Four men and six women are to be seated along a round table. The number of cases where every two men are separated by to women is

Options

A. 3420
B. 43200
C. 1710
D. None of these

Answer

To solve the problem of seating 4 men and 6 women around a round table such that no two men sit next to each other, we can follow these steps:

### Step 1: Arrange the Women
First, we need to arrange the 6 women around the round table. The number of ways to arrange \( n \) people around a round table is given by \( (n-1)! \). Therefore, for 6 women, the number of arrangements is:

\[ (6-1)! = 5! = 120 \]

**Hint:** Remember that in circular permutations, we fix one person to eliminate identical arrangements due to rotation.

### Step 2: Identify Spaces for Men
Once the women are seated, we can visualize the arrangement. With 6 women seated, there will be 6 gaps (spaces) between them where the men can be seated. This can be represented as:

- W1 _ W2 _ W3 _ W4 _ W5 _ W6 _

Here, each underscore (_) represents a gap where a man can be seated.

**Hint:** Count the gaps created by the arrangement of women; these are the only places where men can sit.

### Step 3: Choose Spaces for Men
We need to choose 4 out of the 6 available gaps to place the men. The number of ways to choose 4 gaps from 6 is given by the combination formula \( \binom{n}{r} \):

\[ \binom{6}{4} = \binom{6}{2} = 15 \]

**Hint:** Use the combination formula to select spaces where the men will sit, ensuring that they are separated by women.

### Step 4: Arrange the Men
After selecting the gaps, we need to arrange the 4 men in the chosen gaps. The number of ways to arrange 4 men is given by \( 4! \):

\[ 4! = 24 \]

**Hint:** Remember that the arrangement of distinct individuals (men) is calculated using factorial.

### Step 5: Calculate Total Arrangements
Now, we can find the total number of arrangements by multiplying the arrangements of women, the selection of gaps, and the arrangements of men:

\[ \text{Total arrangements} = \text{(Arrangements of Women)} \times \text{(Ways to choose gaps)} \times \text{(Arrangements of Men)} \]

Substituting the values we calculated:

\[ \text{Total arrangements} = 5! \times \binom{6}{4} \times 4! = 120 \times 15 \times 24 \]

### Step 6: Perform the Calculation
Now we compute the total:

\[ 120 \times 15 = 1800 \]
\[ 1800 \times 24 = 43200 \]

Thus, the total number of arrangements where every two men are separated by women is:

\[ \boxed{43200} \]