8. Twelve balls are distributed randomly among three boxes. What is the probability that the first box will contain 3 balls?

We are tasked with finding the probability that the first box will contain exactly 3 balls when 12 balls are randomly distributed among three boxes. Here's a step-by-step approach:

### Step 1: Total possible outcomes
Each of the 12 balls can be placed in any of the 3 boxes. Therefore, the total number of possible outcomes is given by:

\[
3^{12}
\]
This represents all the possible ways of distributing 12 balls into 3 boxes.

### Step 2: Favorable outcomes
We want to count the number of favorable outcomes where exactly 3 balls are in the first box, and the remaining 9 balls are distributed between the second and third boxes.

1. First, select 3 balls to be placed in the first box. The number of ways to choose 3 balls from 12 is given by the combination:
\[
\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]

2. Now, the remaining 9 balls need to be placed in the second and third boxes. Each ball has 2 choices (it can go into either the second or third box), so there are \(2^9\) ways to distribute these 9 balls.

Thus, the total number of favorable outcomes is:
\[
220 \times 2^9
\]

### Step 3: Probability calculation
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. So the probability that the first box contains exactly 3 balls is:

\[
\text{Probability} = \frac{220 \times 2^9}{3^{12}}
\]

Calculating the powers and simplifying:

\[
2^9 = 512 \quad \text{and} \quad 3^{12} = 531441
\]

Thus, the probability is:

\[
\text{Probability} = \frac{220 \times 512}{531441} = \frac{112640}{531441} \approx 0.2119
\]

So, the probability that the first box will contain exactly 3 balls is approximately **0.2119**, or about **21.19%**.

---

Would you like a more detailed breakdown of any of these steps or have any questions?

Here are some related questions you can explore:

1. What is the probability that the first box will contain exactly 4 balls?
2. What is the probability that the second box will contain exactly 5 balls?
3. What is the probability that one of the boxes will contain all 12 balls?
4. How does the probability change if there are 4 boxes instead of 3?
5. What is the probability that the first box contains 0 balls?

**Tip**: When dealing with combinations, remember that \(\binom{n}{k}\) represents the number of ways to choose \(k\) objects from \(n\) without regard to order.

Twelve balls are distributed randomly among three boxes. What is the probability that the first box will contain 3 balls?

Explore the probability of the first box containing exactly 3 balls when 12 balls are randomly distributed among 3 boxes. This problem covers combinatorics and probability, using key formulas such as the combination formula and total outcomes. Suitable for high school students in Grades 10-12.

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