Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4, and then the men select the chairs from amongst the remaining. The number of possible arrangements is:

Options

A. `""^(4)C_(3)xx""^(4)C_(2)`
B. `""^(4)C_(2)xx""^(4)P_(3)`
C. `""^(4)P_(2)xx""^(4)P_(3)`
D. `""^(4)P_(2)xx""^(6)P_(3)`

Answer

To solve the problem, we need to follow the steps outlined below:

### Step 1: Women Choosing Chairs
The two women can choose chairs from the first four chairs (1, 2, 3, 4). We need to select 2 chairs out of these 4 for the women.

The number of ways to choose 2 chairs from 4 can be calculated using the combination formula:
\[ \text{Number of ways} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Here, \( n = 4 \) and \( r = 2 \):
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 2: Arranging Women in the Chosen Chairs
After choosing the chairs, the two women can occupy the selected chairs in different arrangements. The number of ways to arrange 2 women in 2 chosen chairs is given by:
\[ 2! = 2 \times 1 = 2 \]

### Step 3: Men Choosing Chairs
After the women have chosen their chairs, there will be 2 chairs occupied by women and 2 chairs left from the first four. This means there are 4 chairs remaining for the men to choose from (the 2 remaining from the first four and the 4 from chairs 5 to 8). Therefore, the men can choose from 6 available chairs.

The number of ways to choose 3 chairs from these 6 is:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 4: Arranging Men in the Chosen Chairs
After the men have chosen their chairs, they can occupy the selected chairs in different arrangements. The number of ways to arrange 3 men in 3 chosen chairs is given by:
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Final Calculation
Now, we can calculate the total number of arrangements by multiplying the results from all the steps:
\[ \text{Total arrangements} = (\text{Ways to choose chairs for women}) \times (\text{Ways to arrange women}) \times (\text{Ways to choose chairs for men}) \times (\text{Ways to arrange men}) \]
\[ = 6 \times 2 \times 20 \times 6 \]
\[ = 6 \times 2 = 12 \]
\[ 12 \times 20 = 240 \]
\[ 240 \times 6 = 1440 \]

Thus, the total number of possible arrangements is **1440**.