If there are n student and r prizes `(r lt n)` They can be given away
in `n^r` ways when a student can receive any number of prizes. True or False.

Answer

To determine whether the statement "If there are n students and r prizes (r < n), they can be given away in n^r ways when a student can receive any number of prizes" is true or false, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have `n` students and `r` prizes.
- The condition states that `r < n`, meaning there are more students than prizes.
- Each student can receive any number of prizes, including none.

2. **Distribution of Prizes**:
- For each prize, we have `n` choices (since any of the `n` students can receive it).
- This means that for the first prize, there are `n` possible students who can receive it.

3. **Repeating for Each Prize**:
- The same logic applies to each of the `r` prizes.
- For the second prize, again, there are `n` choices of students.
- This continues for all `r` prizes.

4. **Calculating Total Ways**:
- Since the choices for each prize are independent of each other, we multiply the number of choices for each prize.
- Therefore, the total number of ways to distribute the prizes is:
\[ n \times n \times n \times \ldots \text{ (r times)} = n^r \]

5. **Conclusion**:
- The statement claims that the number of ways to distribute the prizes is `n^r`, which we have shown to be true based on our calculations.
- Thus, the statement is **True**.