A gentleman has 6 friends to invite. In how many ways, can he send invitation cards to them, if he has three servants to carry the cards?

Answer

To solve the problem of how many ways a gentleman can send invitation cards to his 6 friends using 3 servants, we can break it down step by step.

### Step-by-Step Solution:

1. **Identify the Variables**:
- The gentleman has 6 friends (let's denote them as F1, F2, F3, F4, F5, F6).
- He has 3 different servants (which we can think of as 3 different cards).

2. **Understanding the Choices**:
- Each friend can receive an invitation from any of the 3 servants. This means for each friend, there are 3 choices (servant 1, servant 2, or servant 3).

3. **Calculate the Total Choices for Each Friend**:
- For Friend F1, there are 3 choices.
- For Friend F2, there are also 3 choices.
- This pattern continues for all 6 friends.

4. **Use the Multiplication Principle**:
- Since the choice of servant for each friend is independent of the others, we can multiply the number of choices together.
- Therefore, the total number of ways to send invitations to all 6 friends is:
\[ 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \]

5. **Calculate \(3^6\)**:
- We can calculate \(3^6\) as follows:
\[ 3^6 = 729 \]

6. **Conclusion**:
- Thus, the total number of ways the gentleman can send invitation cards to his 6 friends using 3 servants is **729**.