To determine how many ways the prizes can be allocated among athletes, we use permutations because the order of finishing matters (1st, 2nd, 3rd, etc.).
Question 1: 20 total athletes enter the race.
We need to choose 5 athletes from 20 and arrange them in order (since the positions are distinct). The number of ways to do this is given by the permutation formula:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
For our case, \( n = 20 \) and \( r = 5 \):
\[ P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!} = 20 \times 19 \times 18 \times 17 \times 16 \]
Calculating it step-by-step:
- \( 20 \times 19 = 380 \)
- \( 380 \times 18 = 6840 \)
- \( 6840 \times 17 = 116280 \)
- \( 116280 \times 16 = 1860480 \)
So, the number of ways to allocate prizes among 20 athletes is 1,860,480.
Question 2: If only 10 people enter the race.
Using the same permutation formula with \( n = 10 \) and \( r = 5 \):
\[ P(10, 5) = \frac{10!}{(10-5)!} = \frac{10!}{5!} = 10 \times 9 \times 8 \times 7 \times 6 \]
Calculating it step-by-step:
- \( 10 \times 9 = 90 \)
- \( 90 \times 8 = 720 \)
- \( 720 \times 7 = 5040 \)
- \( 5040 \times 6 = 30240 \)
So, the number of ways to allocate prizes among 10 athletes is 30240.
To summarize:
- For 20 athletes: 1,860,480
- For 10 athletes: 30240