In how many ways, 12 balls can be divided between 2 boys, one receiving 5 and the other 7 balls?

Options

A. 1784
B. 1584
C. 1854
D. 1560

Answer

To solve the problem of how many ways 12 balls can be divided between 2 boys, where one boy receives 5 balls and the other receives 7 balls, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Balls**:
We have a total of 12 balls.

2. **Determine the Distribution**:
One boy (let's call him Boy A) will receive 5 balls, and the other boy (Boy B) will receive 7 balls.

3. **Calculate the Number of Ways to Choose Balls for Boy A**:
The number of ways to choose 5 balls out of 12 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 12 \) and \( r = 5 \):
\[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = \frac{12!}{5! \cdot 7!} \]

4. **Simplify the Combination**:
We can simplify \( \frac{12!}{5! \cdot 7!} \):
\[ \binom{12}{5} = \frac{12 \times 11 \times 10 \times 9 \times 8}{5 \times 4 \times 3 \times 2 \times 1} \]

5. **Calculate the Numerator**:
Calculate the product in the numerator:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]
\[ 11880 \times 8 = 95040 \]

6. **Calculate the Denominator**:
Calculate \( 5! \):
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

7. **Final Calculation**:
Now divide the numerator by the denominator:
\[ \frac{95040}{120} = 792 \]

8. **Consider Both Cases**:
Since Boy A can receive either 5 balls or 7 balls, we multiply the result by 2 (for the two different distributions):
\[ 792 \times 2 = 1584 \]

### Final Answer:
Thus, the total number of ways to distribute the 12 balls between the two boys is **1584**.