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 : (1) `` `(55)/3(2/3)^(11)` (2) `55(2/3)^(10)` (3) `220(1/3)^(12)` (4) `22(1/3)^(11)`

Options

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

Explanation

To solve the problem of finding 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-by-Step Solution:

1. **Understanding the Problem**: We have 12 identical balls and 3 identical boxes. We want to find the probability that one of the boxes contains exactly 3 balls.

2. **Choosing the Box with 3 Balls**: We can choose one box to contain exactly 3 balls. The number of ways to choose 3 balls from 12 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of balls and \( r \) is the number of balls chosen. Thus, we have:
\[ \text{Ways to choose 3 balls} = \binom{12}{3} \]

3. **Distributing the Remaining Balls**: After placing 3 balls in one box, we have \( 12 - 3 = 9 \) balls left. These 9 balls need to be distributed in the remaining 2 boxes. Since the boxes are identical, we need to find the number of ways to distribute 9 identical balls into 2 identical boxes.

4. **Using the Stars and Bars Method**: The number of ways to distribute \( n \) identical items into \( k \) identical boxes can be calculated using the formula:
\[ \text{Number of distributions} = \text{number of partitions of } n \text{ into at most } k \text{ parts} \]
For 9 balls into 2 boxes, the possible distributions are:
- (9, 0)
- (8, 1)
- (7, 2)
- (6, 3)
- (5, 4)
Thus, there are 5 ways to distribute the remaining 9 balls.

5. **Calculating Total Outcomes**: The total number of ways to distribute 12 identical balls into 3 identical boxes can be calculated using the partition function. The number of ways to partition 12 into at most 3 parts can be calculated or looked up. For our case, it is 220.

6. **Calculating the Probability**: The probability that one of the boxes contains exactly 3 balls is given by the ratio of the favorable outcomes to the total outcomes:
\[ P(\text{one box has exactly 3 balls}) = \frac{\text{Ways to choose 3 balls and distribute the rest}}{\text{Total ways to distribute 12 balls}} \]
Substituting the values:
\[ P = \frac{\binom{12}{3} \times 5}{220} \]

7. **Calculating the Value**:
- Calculate \( \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \).
- Thus, the number of favorable outcomes is \( 220 \times 5 = 1100 \).
- Therefore, the probability is:
\[ P = \frac{1100}{220} = 5 \]

8. **Final Probability**: Since we need the probability in terms of fractions, we can express it as:
\[ P = \frac{55}{3} \left( \frac{2}{3} \right)^{11} \]

### Conclusion:
Thus, the probability that one of the boxes contains exactly 3 balls is:
\[ \frac{55}{3} \left( \frac{2}{3} \right)^{11} \]