In how many ways can 4 prizes be given to 3 boys when a boy is eligible for all prizes?

Explanation

To solve the problem of distributing 4 prizes to 3 boys where each boy is eligible for all prizes, we will use the concept of permutations with repetition. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have 4 distinct prizes and 3 boys. Each prize can go to any of the 3 boys.

### Step 2: Determine Choices for Each Prize
For each of the 4 prizes, we have 3 choices (one for each boy).

### Step 3: Calculate Total Combinations
Since the choice of which boy receives each prize is independent, we can multiply the number of choices for each prize:
- For Prize 1, there are 3 choices (Boy 1, Boy 2, Boy 3).
- For Prize 2, there are again 3 choices.
- For Prize 3, there are again 3 choices.
- For Prize 4, there are again 3 choices.

Thus, the total number of ways to distribute the prizes is:
\[ 3 \times 3 \times 3 \times 3 = 3^4 \]

### Step 4: Calculate \(3^4\)
Now, we calculate \(3^4\):
\[ 3^4 = 81 \]

### Conclusion
Therefore, the total number of ways to distribute 4 prizes to 3 boys is **81**.

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