Sixteen men compete with one another in running, swimming and riding. How many prize lists could be made if there were altogether 6 prizes of different values , one for running 2 for swimming and 3 for riding?

Answer

To solve the problem of how many prize lists could be made for the competitions in running, swimming, and riding, we will break it down step by step.

### Step-by-Step Solution:

1. **Identify the Prizes and Competitions**:
- There are 6 prizes in total:
- 1 prize for running
- 2 prizes for swimming
- 3 prizes for riding

2. **Calculate the Number of Ways to Award the Running Prize**:
- For the running prize, since there is only 1 prize and 16 competitors, any one of the 16 men can win it.
- Therefore, the number of ways to award the running prize = 16.

\[ \text{Ways for running} = 16 \]

3. **Calculate the Number of Ways to Award the Swimming Prizes**:
- For the swimming prizes, there are 2 prizes to be awarded to 16 competitors.
- The first swimming prize can be awarded to any of the 16 men.
- After awarding the first swimming prize, 15 men remain eligible for the second swimming prize.
- Therefore, the number of ways to award the swimming prizes = 16 (for the first prize) × 15 (for the second prize).

\[ \text{Ways for swimming} = 16 \times 15 \]

4. **Calculate the Number of Ways to Award the Riding Prizes**:
- For the riding prizes, there are 3 prizes to be awarded.
- The first riding prize can be awarded to any of the 16 men.
- The second riding prize can be awarded to any of the remaining 15 men.
- The third riding prize can be awarded to any of the remaining 14 men.
- Therefore, the number of ways to award the riding prizes = 16 (for the first prize) × 15 (for the second prize) × 14 (for the third prize).

\[ \text{Ways for riding} = 16 \times 15 \times 14 \]

5. **Combine All the Ways**:
- Since the events (running, swimming, and riding) are independent, we multiply the number of ways for each event to find the total number of prize lists.

\[ \text{Total Ways} = \text{Ways for running} \times \text{Ways for swimming} \times \text{Ways for riding} \]

\[ \text{Total Ways} = 16 \times (16 \times 15) \times (16 \times 15 \times 14) \]

\[ \text{Total Ways} = 16 \times 16 \times 15 \times 16 \times 15 \times 14 \]

6. **Final Calculation**:
- Now we can compute the total number of ways.

\[ \text{Total Ways} = 16^3 \times 15^2 \times 14 \]

### Final Answer:
The total number of prize lists that could be made is \( 16^3 \times 15^2 \times 14 \).