If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to:

Options

A. 47
B. 53
C. 51
D. 43

Answer

To solve the problem of distributing 5 different employees into 4 indistinguishable offices, we can use the concept of partitions of a set. The problem can be approached by considering the different ways to partition the employees into groups, where the groups represent the offices.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the number of ways to distribute 5 different employees into 4 indistinguishable offices. Each office can hold any number of employees, including zero.

2. **Using Stirling Numbers of the Second Kind**: The number of ways to partition \( n \) distinct objects into \( k \) indistinguishable boxes (offices) is given by the Stirling number of the second kind, denoted as \( S(n, k) \). Here, \( n = 5 \) and \( k = 4 \).

3. **Calculating \( S(5, 4) \)**:
- The Stirling number \( S(5, 4) \) represents the number of ways to partition 5 employees into 4 non-empty groups. This can be calculated using the recurrence relation:
\[ S(n, k) = k \cdot S(n-1, k) + S(n-1, k-1) \]
- We need to calculate \( S(5, 4) \):
- \( S(4, 4) = 1 \) (all employees in separate offices)
- \( S(4, 3) = 6 \) (one office has 2 employees, and the others have 1 each)
- \( S(3, 3) = 1 \)
- \( S(3, 2) = 3 \)
- \( S(2, 2) = 1 \)
- \( S(2, 1) = 1 \)
- \( S(1, 1) = 1 \)

4. **Calculating \( S(5, 4) \)**:
- Using the recurrence relation:
\[ S(5, 4) = 4 \cdot S(4, 4) + S(4, 3) = 4 \cdot 1 + 6 = 10 \]

5. **Calculating \( S(5, 3) \)**:
- We also need \( S(5, 3) \) since it represents the case where one office is empty:
\[ S(5, 3) = 3 \cdot S(4, 3) + S(4, 2) = 3 \cdot 6 + 7 = 18 + 7 = 25 \]

6. **Total Ways**:
- Now we can calculate the total number of distributions:
- \( S(5, 4) + S(5, 3) = 10 + 25 = 35 \)

7. **Final Calculation**:
- The total number of ways to distribute 5 employees into 4 indistinguishable offices is \( 35 \).

### Conclusion:
The total number of ways \( n \) is equal to **35**.