Out of 8 sailors on a boat, 3 can work only on one particular side and 2 only on the other side. Find the number of ways in which the ways in which the sailors can be arranged on the boat.

Explanation

To solve the problem step by step, we will analyze how to arrange the sailors on the boat given the constraints.

### Step 1: Understand the constraints
We have a total of 8 sailors:
- 3 sailors can only work on one particular side (let's say the left side).
- 2 sailors can only work on the other side (let's say the right side).
- This leaves us with 3 sailors who can work on either side.

### Step 2: Determine the arrangement of sailors
Since we need to have 4 sailors on each side for balance:
- The left side will have the 3 fixed sailors plus 1 from the remaining 3 sailors.
- The right side will have the 2 fixed sailors plus 2 from the remaining 3 sailors.

### Step 3: Choose sailors for each side
We need to select:
- 1 sailor from the 3 remaining sailors to join the left side.
- 2 sailors from the remaining 2 sailors to join the right side.

The number of ways to choose 1 sailor from 3 for the left side is given by \( \binom{3}{1} \).
The number of ways to choose 2 sailors from the remaining 2 for the right side is given by \( \binom{2}{2} \).

Calculating these:
\[ \binom{3}{1} = 3 \]
\[ \binom{2}{2} = 1 \]

### Step 4: Calculate the total combinations
Now, we multiply the number of ways to choose the sailors:
\[ \text{Total ways to choose sailors} = \binom{3}{1} \times \binom{2}{2} = 3 \times 1 = 3 \]

### Step 5: Arrange the sailors on each side
Now, we need to arrange the sailors on both sides:
- The left side has 4 sailors (3 fixed + 1 chosen).
- The right side has 4 sailors (2 fixed + 2 chosen).

The number of arrangements for each side is given by \( 4! \) (factorial of 4).

Calculating \( 4! \):
\[ 4! = 24 \]

### Step 6: Calculate the total arrangements
Now, we multiply the number of ways to choose the sailors by the arrangements on each side:
\[ \text{Total arrangements} = \text{Ways to choose sailors} \times \text{Arrangements on left side} \times \text{Arrangements on right side} \]
\[ = 3 \times 24 \times 24 \]
\[ = 3 \times 576 = 1728 \]

### Final Answer
The total number of ways in which the sailors can be arranged on the boat is **1728**.

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