The total number of ways in which 8 men and 6 women can be arranged in a line so that no 2 women are together is

Options

A. 48
B. `""^(8)p_(8).""_(9) p_(6)`
C. ` 7! *^(8)p_(6)`
D. ` ""^(8) c_(8) .""^(9)c_(8)`

Answer

To find the total number of ways in which 8 men and 6 women can be arranged in a line such that no two women are together, we can follow these steps:

### Step-by-Step Solution:

1. **Arrange the Men**:
First, we arrange the 8 men in a line. The number of ways to arrange 8 men is given by \(8!\) (8 factorial).

\[ \text{Ways to arrange men} = 8! = 40320 \]

**Hint**: Remember that the factorial of a number \(n\) (denoted \(n!\)) is the product of all positive integers up to \(n\).

2. **Identify Gaps for Women**:
Once the men are arranged, we can identify the gaps where the women can be placed. With 8 men, there are 9 possible gaps (one before each man, one after the last man).

\[ \text{Gaps} = 9 \]

**Hint**: Visualize the arrangement of men as M M M M M M M M, where the underscores represent gaps: _ M _ M _ M _ M _ M _ M _ M _.

3. **Select Gaps for Women**:
We need to choose 6 out of these 9 gaps to place the women. The number of ways to choose 6 gaps from 9 is given by the combination formula \( \binom{n}{r} \), which is \( \binom{9}{6} \).

\[ \text{Ways to choose gaps} = \binom{9}{6} = \binom{9}{3} = \frac{9!}{6! \cdot 3!} = 84 \]

**Hint**: Use the property of combinations that \( \binom{n}{r} = \binom{n}{n-r} \) to simplify calculations.

4. **Arrange the Women**:
After selecting the gaps, we need to arrange the 6 women in those selected gaps. The number of ways to arrange 6 women is given by \(6!\).

\[ \text{Ways to arrange women} = 6! = 720 \]

**Hint**: Just like with the men, the arrangement of women also uses factorial.

5. **Calculate Total Arrangements**:
Finally, we multiply the number of ways to arrange the men, choose the gaps, and arrange the women to get the total number of arrangements.

\[ \text{Total arrangements} = (8!) \times \left(\binom{9}{6}\right) \times (6!) \]
\[ = 40320 \times 84 \times 720 \]

**Hint**: Ensure to multiply all components together to find the final result.

6. **Final Calculation**:
Now, let's compute the total arrangements:

\[ 40320 \times 84 = 3386880 \]
\[ 3386880 \times 720 = 2432908800 \]

Thus, the total number of ways in which 8 men and 6 women can be arranged in a line so that no two women are together is:

\[ \text{Total Ways} = 2432908800 \]

### Final Answer:
The total number of ways in which 8 men and 6 women can be arranged in a line so that no 2 women are together is **2432908800**.