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 th chairs from amongst the remaining. The number of possible arrangements is a.`^6C_3xx^4C_2` b. `^4P_2xx^4P_3` c. `^4C_2xx^4P_3` d. none of these

Explanation

To solve the problem step by step, we will analyze the choices made by the women and men in detail.

### Step 1: Women Choosing Chairs
The two women will choose chairs from the first four chairs (1, 2, 3, 4). The number of ways to choose 2 chairs from these 4 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Ways for women to choose chairs} = \binom{4}{2} \]

### Step 2: Arranging the Women
After choosing the chairs, the two women can occupy the selected chairs in different arrangements. The number of arrangements of 2 women in 2 chosen chairs is given by \( 2! \) (factorial of the number of women).

\[ \text{Ways to arrange women} = 2! \]

### Step 3: Remaining Chairs for Men
After the women have chosen their chairs, 2 chairs are occupied, leaving 6 chairs available (chairs 5, 6, 7, and 8 plus the 2 unchosen chairs from 1 to 4). The men will choose from these remaining 6 chairs.

### Step 4: Men Choosing Chairs
The number of ways the 3 men can choose 3 chairs from the remaining 6 chairs is given by the combination formula \( \binom{6}{3} \).

\[ \text{Ways for men to choose chairs} = \binom{6}{3} \]

### Step 5: Arranging the Men
After the men have chosen their chairs, they can occupy the selected chairs in different arrangements. The number of arrangements of 3 men in 3 chosen chairs is given by \( 3! \).

\[ \text{Ways to arrange men} = 3! \]

### Step 6: Total Arrangements
Now, we can find the total number of arrangements by multiplying the number of ways the women can choose and arrange their chairs with the number of ways the men can choose and arrange their chairs.

\[ \text{Total arrangements} = \binom{4}{2} \times 2! \times \binom{6}{3} \times 3! \]

### Step 7: Calculation
Now we can compute the values:

1. \( \binom{4}{2} = 6 \)
2. \( 2! = 2 \)
3. \( \binom{6}{3} = 20 \)
4. \( 3! = 6 \)

Putting it all together:

\[ \text{Total arrangements} = 6 \times 2 \times 20 \times 6 = 1440 \]

### Conclusion
Since none of the provided options match this calculation, the answer is:

**d. none of these**

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