Find the probability of placing n balls into n numbered boxes so that exactly
one box remains empty.


To find the probability of placing \( n \) balls into \( n \) numbered boxes such that exactly one box remains empty, we need to approach the problem in two steps:

1. **Calculate the number of favorable outcomes (i.e., exactly one box is empty):**
   - Choose one box to remain empty: There are \( \binom{n}{1} = n \) ways to choose 1 box to remain empty.
   - Distribute the \( n \) balls into the remaining \( n - 1 \) boxes: Each ball can go into one of the remaining \( n - 1 \) boxes. Thus, there are \( (n - 1)^n \) ways to distribute \( n \) balls into \( n - 1 \) boxes.

2. **Calculate the total number of outcomes (i.e., no restriction on the number of balls in each box):**
   - Distribute \( n \) balls into \( n \) boxes: Each ball can go into any of the \( n \) boxes, so there are \( n^n \) total ways to place the balls into the boxes.

3. **The probability** is the ratio of favorable outcomes to the total number of outcomes:
   \[
   P(\text{exactly one empty box}) = \frac{n \cdot (n - 1)^n}{n^n}
   \]
   Simplifying this expression:
   \[
   P(\text{exactly one empty box}) = \frac{n \cdot (n - 1)^n}{n^n} = \left(\frac{n-1}{n}\right)^n
   \]

### Final Answer:
The probability of placing \( n \) balls into \( n \) numbered boxes such that exactly one box remains empty is:
\[
P = \left(\frac{n-1}{n}\right)^n
\]

Would you like further details on any part of the explanation? 

Here are 5 relative questions for further exploration:

1. How does this probability change as \( n \) approaches infinity?
2. What is the probability of exactly two boxes being empty?
3. How does this formula change if the boxes are not numbered?
4. What is the probability that no box is empty?
5. How does this scenario relate to the occupancy problem in probability theory?

**Tip:** In combinatorics, breaking down a problem into favorable and total outcomes is often the most efficient way to compute probabilities.

Find the probability of placing n balls into n numbered boxes so that exactly one box remains empty.

Explore the probability of placing n balls into n numbered boxes with exactly one empty box. This detailed solution covers key probability and combinatorics concepts such as the binomial coefficient and the occupancy problem. Suitable for advanced high school and undergraduate students.

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