The number of ways in which 12 balls can be divided between two friends, one receiving 8 and the other 4, is

Options

A. `(12!)/(8!4!)`
B. `(12!12!)/(8!4!)`
C. `(12!)/(8!4!2!)`
D. none of these

Answer

To solve the problem of dividing 12 balls between two friends, where one receives 8 balls and the other receives 4 balls, we can follow these steps:

### Step-by-Step Solution

1. **Choosing Balls for the First Friend**:
We need to select 8 balls out of the 12 available. This can be represented mathematically using combinations:
\[ \text{Number of ways to choose 8 balls from 12} = \binom{12}{8} \]

2. **Choosing Balls for the Second Friend**:
After selecting 8 balls for the first friend, the remaining 4 balls automatically go to the second friend. The number of ways to choose these 4 balls from the remaining 4 is:
\[ \text{Number of ways to choose 4 balls from 4} = \binom{4}{4} = 1 \]

3. **Considering the Distribution**:
There are two scenarios for the distribution:
- Scenario 1: The first friend gets 8 balls and the second friend gets 4 balls.
- Scenario 2: The first friend gets 4 balls and the second friend gets 8 balls.

Since these two scenarios are mutually exclusive, we can add the number of ways for each scenario. Thus, we multiply the combinations by 2:
\[ \text{Total ways} = \binom{12}{8} \times \binom{4}{4} \times 2 \]

4. **Calculating the Combinations**:
We can calculate the combinations:
\[ \binom{12}{8} = \frac{12!}{8! \cdot (12-8)!} = \frac{12!}{8! \cdot 4!} \]
And since \(\binom{4}{4} = 1\), we can simplify our total ways to:
\[ \text{Total ways} = \binom{12}{8} \times 1 \times 2 = 2 \times \frac{12!}{8! \cdot 4!} \]

5. **Final Expression**:
Therefore, the total number of ways to divide the 12 balls between the two friends is:
\[ \text{Total ways} = \frac{12! \times 2}{8! \times 4!} \]

### Final Answer
The number of ways in which 12 balls can be divided between two friends, one receiving 8 and the other 4, is:
\[ \frac{12! \times 2}{8! \times 4!} \]