A boat's crew consists of 8 men, 3 of whom can only row on one side, 2 only on the other. The number of ways the crew can be arranged.

Options

A. 1728
B. `14 xx 9 !`
C. `""^(12)C_(8)`
D. `""^(9)C_(5)`

Answer

To solve the problem of arranging the crew of a boat consisting of 8 men, where 3 can only row on one side and 2 can only row on the other side, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Crew Members**:
- Let’s denote the crew members as follows:
- A, B, C (can only row on side 1)
- D, E (can only row on side 2)
- F, G, H (can row on either side)

2. **Determine Fixed Positions**:
- Since 3 men (A, B, C) can only row on side 1, they must be placed there.
- Since 2 men (D, E) can only row on side 2, they must be placed there.

3. **Calculate Remaining Positions**:
- Each side needs 4 rowers.
- Side 1 already has 3 rowers (A, B, C), so we need 1 more rower from F, G, H.
- Side 2 already has 2 rowers (D, E), so we need 2 more rowers from F, G, H.

4. **Choose Rowers for Each Side**:
- We need to choose 1 rower from F, G, H to join A, B, C on side 1. This can be done in \( \binom{3}{1} \) ways.
- The remaining 2 rowers will automatically go to side 2.

5. **Calculate Combinations**:
- The number of ways to choose 1 rower for side 1 from F, G, H is:
\[ \binom{3}{1} = 3 \]
- The remaining 2 rowers will go to side 2, which is determined automatically.

6. **Arrange Rowers on Each Side**:
- The rowers on side 1 (A, B, C, and the chosen rower) can be arranged among themselves in \( 4! \) ways.
- The rowers on side 2 (D, E, and the two chosen rowers) can be arranged among themselves in \( 3! \) ways.

7. **Total Arrangements**:
- Therefore, the total number of arrangements is given by:
\[ \text{Total Ways} = \binom{3}{1} \times 4! \times 3! \]
- Substituting the values:
\[ \text{Total Ways} = 3 \times 24 \times 6 = 432 \]

### Final Answer:
The total number of ways the crew can be arranged is **432**.