The crew of an 8-member rowing team is to be chosen from 12 men (`M_1, M_2, ...., M_(12)`) and 8 women `(W_1, W_2, ...., W_8)`. There have to be 4 people on each side with at least one woman on each side. Further it is also known that on the right side of the boat (while going forward) `W_1` and `M_1` must be selected while on the left side of the boat `M_2, M_3` and M10 must be selected. What is the number of ways in which the rowing team can be arranged?

Options

A. `1368 xx 4! xx 4!`
B. `1274 xx 4! xx 4!`
C. `1120 xx 4 ! xx 4!`
D. `728 xx 4! xx 4!`

Answer

To solve the problem of arranging the rowing team, we will break it down step by step.

### Step 1: Understand the Composition of the Team
We have a total of 12 men and 8 women. We need to form a rowing team of 8 members, with 4 members on each side of the boat. Each side must have at least one woman.

### Step 2: Assign Fixed Members
We know that:
- On the right side, `W_1` (a woman) and `M_1` (a man) must be selected.
- On the left side, `M_2`, `M_3`, and `M_10` (all men) must be selected.

### Step 3: Determine Remaining Members
After assigning the fixed members:
- **Right Side**: We have already selected `W_1` and `M_1`, leaving us to select 2 more members.
- **Left Side**: We have already selected `M_2`, `M_3`, and `M_10`, leaving us to select 1 more member.

### Step 4: Count Available Members
- Remaining members after selecting the fixed ones:
- Men left: 12 total - 4 selected = 8 men left (M_4, M_5, M_6, M_7, M_8, M_9, M_11, M_12)
- Women left: 8 total - 1 selected = 7 women left (W_2, W_3, W_4, W_5, W_6, W_7, W_8)

### Step 5: Select Members for Each Side
1. **Right Side Selection**:
- We need to select 2 more members from the remaining 8 men and 7 women, ensuring at least one woman is included.
- We can choose:
- 1 woman and 1 man, or
- 2 women

2. **Left Side Selection**:
- We need to select 1 more member from the remaining 8 men and 7 women, ensuring at least one woman is included.
- We can choose:
- 1 woman, or
- 1 man (but since we need at least one woman on this side, we will select a woman).

### Step 6: Calculate Combinations
1. **Right Side Combinations**:
- **Case 1**: 1 woman and 1 man
- Choose 1 woman from 7: \( \binom{7}{1} = 7 \)
- Choose 1 man from 8: \( \binom{8}{1} = 8 \)
- Total for this case: \( 7 \times 8 = 56 \)

- **Case 2**: 2 women
- Choose 2 women from 7: \( \binom{7}{2} = 21 \)
- Total for this case: 21

- Total combinations for the right side: \( 56 + 21 = 77 \)

2. **Left Side Combinations**:
- We need to choose 1 woman from the remaining 6 women (since `W_1` is already on the right side):
- Choose 1 woman from 6: \( \binom{6}{1} = 6 \)

### Step 7: Calculate Arrangements
- The arrangements for each side:
- Right Side: 4 members can be arranged in \( 4! = 24 \) ways.
- Left Side: 4 members can be arranged in \( 4! = 24 \) ways.

### Step 8: Total Combinations
The total number of ways to arrange the rowing team is given by:
\[ \text{Total Ways} = (\text{Ways to choose for Right Side}) \times (\text{Ways to choose for Left Side}) \times (\text{Arrangements for Right Side}) \times (\text{Arrangements for Left Side}) \]
\[ = (77) \times (6) \times (24) \times (24) \]

### Step 9: Final Calculation
Calculating the total:
\[ = 77 \times 6 \times 24 \times 24 = 77 \times 6 \times 576 = 77 \times 3456 = 265632 \]

Thus, the total number of ways to arrange the rowing team is **265632**.