If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is

Options

A. `55/3(2/3)^(11)`
B. `55(2/3)^(10)`
C. `220(1/3)^(12)`
D. `22(11/3)^(11)`

Answer

To find the probability that one of the boxes contains exactly 3 balls when 12 identical balls are placed in 3 identical boxes, we can follow these steps:

### Step 1: Calculate the Total Number of Ways to Distribute the Balls
Each of the 12 identical balls can go into any of the 3 identical boxes. Therefore, the total number of ways to distribute the 12 balls into the 3 boxes is given by:
\[ \text{Total Cases} = 3^{12} \]

### Step 2: Determine the Favorable Cases
We need to find the number of ways to distribute the balls such that exactly one box contains exactly 3 balls.

1. **Choose 3 Balls for One Box**: We can choose 3 balls out of the 12 to place in one box. The number of ways to choose 3 balls from 12 is given by the combination formula:
\[ \text{Ways to choose 3 balls} = \binom{12}{3} \]

2. **Distribute the Remaining Balls**: After placing 3 balls in one box, we have 9 balls left to distribute into the remaining 2 boxes. Since the boxes are identical, each of the remaining 9 balls can go into either of the 2 boxes. Therefore, the number of ways to distribute these 9 balls is:
\[ \text{Ways to distribute 9 balls} = 2^9 \]

3. **Combine the Favorable Cases**: The total number of favorable cases where one box contains exactly 3 balls is:
\[ \text{Favorable Cases} = \binom{12}{3} \times 2^9 \]

### Step 3: Calculate the Probability
The probability \( P \) that one of the boxes contains exactly 3 balls is given by the ratio of the favorable cases to the total cases:
\[ P = \frac{\text{Favorable Cases}}{\text{Total Cases}} = \frac{\binom{12}{3} \times 2^9}{3^{12}} \]

### Step 4: Simplify the Expression
1. Calculate \( \binom{12}{3} \):
\[ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]

2. Calculate \( 2^9 \):
\[ 2^9 = 512 \]

3. Calculate \( 3^{12} \):
\[ 3^{12} = 531441 \]

4. Substitute these values into the probability formula:
\[ P = \frac{220 \times 512}{531441} \]

5. Calculate \( 220 \times 512 = 112640 \):
\[ P = \frac{112640}{531441} \]

### Final Result
Thus, the probability that one of the boxes contains exactly 3 balls is:
\[ P = \frac{112640}{531441} \]