It is required to sent 7 men and 3 women in a row so that women occupy the even places. How many such arrangements are possible?

Options

A. `^(5)C_(3)xx3!xx7!`
B. `^(10)C_(5)xx5!`
C. `^(5)C_(3)xx10!`
D. none of these

Explanation

To solve the problem of arranging 7 men and 3 women in a row such that the women occupy the even places, we can follow these steps:

### Step 1: Identify the positions
In a row of 10 seats (for 7 men and 3 women), the even positions are 2, 4, 6, 8, and 10. This gives us 5 available even positions.

### Step 2: Choose positions for the women
We need to choose 3 out of these 5 even positions for the women. The number of ways to choose 3 positions from 5 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of positions and \( r \) is the number of positions to choose.

\[ \text{Number of ways to choose positions for women} = \binom{5}{3} \]

### Step 3: Arrange the women
Once we have chosen the positions for the women, we need to arrange the 3 women in those chosen positions. The number of ways to arrange 3 women is given by \( 3! \) (3 factorial).

\[ \text{Number of ways to arrange women} = 3! \]

### Step 4: Arrange the men
After placing the women, we have 7 positions left for the men. The number of ways to arrange 7 men in these 7 positions is given by \( 7! \) (7 factorial).

\[ \text{Number of ways to arrange men} = 7! \]

### Step 5: Calculate the total arrangements
The total number of arrangements is the product of the number of ways to choose positions for the women, the arrangements of the women, and the arrangements of the men.

\[ \text{Total arrangements} = \binom{5}{3} \times 3! \times 7! \]

### Step 6: Compute the values
Now, we calculate each part:

1. \( \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \)
2. \( 3! = 6 \)
3. \( 7! = 5040 \)

Now, substituting these values back into the total arrangements formula:

\[ \text{Total arrangements} = 10 \times 6 \times 5040 \]

Calculating this gives:

\[ \text{Total arrangements} = 10 \times 6 = 60 \]
\[ 60 \times 5040 = 302400 \]

Thus, the total number of arrangements is **302400**.

### Final Answer
The total number of arrangements of 7 men and 3 women in a row such that women occupy the even places is **302400**.

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