A man has 3 shirts, 4 trousers and 6 ties. What are the number of ways in which he can dress himself with a combination of all the three?

Options

A. 13
B. 72
C. 13! / 3! 4! 6!
D. 3! 4! 6!

Answer

To solve the problem of how many ways a man can dress himself with a combination of shirts, trousers, and ties, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Choices**:
- The man has 3 shirts.
- The man has 4 trousers.
- The man has 6 ties.

2. **Calculate the Number of Ways to Choose Each Item**:
- For shirts, since he can wear only one shirt at a time, the number of ways to choose a shirt is \(3C1 = 3\).
- For trousers, similarly, the number of ways to choose trousers is \(4C1 = 4\).
- For ties, the number of ways to choose a tie is \(6C1 = 6\).

3. **Combine the Choices**:
- Since the choices are independent (the choice of shirt does not affect the choice of trousers or ties), we multiply the number of ways to choose each item.
- Therefore, the total number of ways to dress himself is:
\[ \text{Total Ways} = (\text{Number of Shirts}) \times (\text{Number of Trousers}) \times (\text{Number of Ties}) = 3 \times 4 \times 6 \]

4. **Calculate the Total**:
- Now, we perform the multiplication:
\[ 3 \times 4 = 12 \]
\[ 12 \times 6 = 72 \]

5. **Final Answer**:
- The total number of ways the man can dress himself is **72**.