To answer the questions about how many ways the prizes can be allocated, we will use the concept of permutations since the order of the winners matters.
Question 1: If 20 total athletes enter the race, how many ways can the prizes be allocated?
We are choosing 5 out of 20 athletes to receive distinct prizes:
- The number of ways to arrange 5 athletes out of 20 can be calculated as \( P(20, 5) \).
The formula for permutations is: \[ P(n, k) = \frac{n!}{(n-k)!} \]
For our case: \[ P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!} = 20 \times 19 \times 18 \times 17 \times 16 \]
Calculating that step by step:
- \( 20 \times 19 = 380 \)
- \( 380 \times 18 = 6,840 \)
- \( 6,840 \times 17 = 116,280 \)
- \( 116,280 \times 16 = 1,860,480 \)
So, the number of ways to allocate the prizes with 20 athletes is 1,860,480.
Question 2: If only 10 people enter the race, how many ways can the prizes be allocated?
Using the same logic: \[ P(10, 5) = \frac{10!}{(10-5)!} = \frac{10!}{5!} = 10 \times 9 \times 8 \times 7 \times 6 \]
Calculating that step by step:
- \( 10 \times 9 = 90 \)
- \( 90 \times 8 = 720 \)
- \( 720 \times 7 = 5,040 \)
- \( 5,040 \times 6 = 30,240 \)
So, the number of ways to allocate the prizes with 10 athletes is 30240.
Final answers: Question 1: 1,860,480 Question 2: 30240