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 calculate both the total number of ways to distribute the balls and the number of favorable outcomes where exactly one box is empty.

### Step 1: Total number of outcomes
The total number of ways to place \( n \) balls into \( n \) boxes without any restriction is:
\[
n^n
\]
since each ball can go into any of the \( n \) boxes.

### Step 2: Favorable outcomes (one empty box)
Now, we need to calculate the number of ways to place \( n \) balls into \( n \) boxes such that exactly one box is empty.

1. **Choose which box remains empty**: There are \( \binom{n}{1} = n \) ways to choose one box to be empty.

2. **Distribute \( n \) balls into \( n-1 \) boxes**: The number of ways to place \( n \) balls into \( n-1 \) boxes such that no box is empty (i.e., each box gets at least one ball) is given by a well-known result called the **Stirling number of the second kind**. Specifically, this is equivalent to placing \( n \) balls into \( n-1 \) boxes with no empty boxes, which is \( (n-1)^{n-1} \) (derived from the inclusion-exclusion principle).

Thus, the number of favorable outcomes is:
\[
n \times (n-1)^{n-1}
\]

### Step 3: Probability
The probability is the ratio of favorable outcomes to the total number of outcomes:
\[
P(\text{one empty box}) = \frac{n \times (n-1)^{n-1}}{n^n}
\]

Simplifying this expression:
\[
P(\text{one empty box}) = \frac{(n-1)^{n-1}}{n^{n-1}}
\]

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

---

Would you like further details or have any questions? Here are 5 related questions you might find interesting:

1. How does the probability change as \( n \) increases?
2. What is the probability of exactly \( k \) boxes remaining empty?
3. How would this probability change if the balls were indistinguishable?
4. Can this problem be generalized to more than \( n \) balls?
5. What is the expected number of empty boxes when \( n \) balls are placed into \( n \) boxes?

**Tip**: When working with probabilities, always ensure the favorable outcomes and total outcomes are clearly defined to avoid errors!

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 such that exactly one box remains empty. This problem covers advanced combinatorics, Stirling numbers, and the inclusion-exclusion principle. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation