There are 5 different caps and 5 different boxes. If at least one cap has to be distributed and the caps have to be arranged such that any box can have a maximum of one cap only, in how many ways can you arrange the caps among 5 boxes?

Options

A. a. 1545
B. b. 600
C. c. 1768
D. d. none of these

Answer

To solve the problem of distributing 5 different caps into 5 different boxes with the condition that at least one cap must be distributed and each box can hold at most one cap, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 5 different caps and 5 different boxes. We need to distribute at least one cap into the boxes, ensuring that no box contains more than one cap.

2. **Cases of Distribution**: Since we can distribute anywhere from 1 to 5 caps, we will consider each case separately:
- Case 1: Distributing 1 cap
- Case 2: Distributing 2 caps
- Case 3: Distributing 3 caps
- Case 4: Distributing 4 caps
- Case 5: Distributing 5 caps

3. **Calculating Each Case**:
- **Case 1 (1 cap)**:
- Choose 1 cap from 5: \( \binom{5}{1} = 5 \)
- Choose 1 box from 5: \( \binom{5}{1} = 5 \)
- Total ways = \( 5 \times 5 = 25 \)

- **Case 2 (2 caps)**:
- Choose 2 caps from 5: \( \binom{5}{2} = 10 \)
- Choose 2 boxes from 5: \( \binom{5}{2} = 10 \)
- Arrange 2 caps in 2 boxes: \( 2! = 2 \)
- Total ways = \( 10 \times 10 \times 2 = 200 \)

- **Case 3 (3 caps)**:
- Choose 3 caps from 5: \( \binom{5}{3} = 10 \)
- Choose 3 boxes from 5: \( \binom{5}{3} = 10 \)
- Arrange 3 caps in 3 boxes: \( 3! = 6 \)
- Total ways = \( 10 \times 10 \times 6 = 600 \)

- **Case 4 (4 caps)**:
- Choose 4 caps from 5: \( \binom{5}{4} = 5 \)
- Choose 4 boxes from 5: \( \binom{5}{4} = 5 \)
- Arrange 4 caps in 4 boxes: \( 4! = 24 \)
- Total ways = \( 5 \times 5 \times 24 = 600 \)

- **Case 5 (5 caps)**:
- Choose all 5 caps: \( \binom{5}{5} = 1 \)
- Choose all 5 boxes: \( \binom{5}{5} = 1 \)
- Arrange 5 caps in 5 boxes: \( 5! = 120 \)
- Total ways = \( 1 \times 1 \times 120 = 120 \)

4. **Summing All Cases**:
- Total arrangements = Case 1 + Case 2 + Case 3 + Case 4 + Case 5
- Total arrangements = \( 25 + 200 + 600 + 600 + 120 = 1545 \)

### Final Answer:
The total number of ways to arrange the caps among the boxes is **1545**.