In how many ways can 8 distinct things be divided among three people such that any one can receive any number of things?

Options

A. `1556`
B. 6561
C. `""^(8)C_2`
D. `""^(7)C_2`

Answer

To solve the problem of how many ways 8 distinct things can be divided among three people such that any one can receive any number of things, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have 8 distinct items and 3 people. Each item can be given to any one of the three people.

2. **Choices for Each Item**: For each of the 8 distinct items, there are 3 choices (it can go to person 1, person 2, or person 3).

3. **Calculating Total Choices**: Since the choice for each item is independent of the others, we can use the multiplication principle of counting. This means we multiply the number of choices for each item together.

- For the first item, there are 3 choices.
- For the second item, there are also 3 choices.
- This continues for all 8 items.

Therefore, the total number of ways to distribute the items is:
\[ 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^8 \]

4. **Calculating \(3^8\)**: Now we need to compute \(3^8\):
\[ 3^8 = 6561 \]

5. **Conclusion**: Thus, the total number of ways to divide 8 distinct things among 3 people is **6561**.

### Final Answer:
The answer is **6561**.

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