To determine the number of ways to allocate the prizes among the top 5 finishers in a race with a total of athletes, we need to calculate the permutations of 20 athletes taken 5 at a time.
Question 1: When there are 20 athletes, we calculate the number of ways to choose and order 5 of them using the permutation formula:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
Where:
- \( n = 20 \) (total athletes)
- \( r = 5 \) (top 5 finishers)
So we have:
\[ P(20, 5) = \frac{20!}{(20 - 5)!} = \frac{20!}{15!} = 20 \times 19 \times 18 \times 17 \times 16 \]
Calculating that step-by-step:
- \( 20 \times 19 = 380 \)
- \( 380 \times 18 = 6840 \)
- \( 6840 \times 17 = 116280 \)
- \( 116280 \times 16 = 1860480 \)
So, there are 1,860,480 ways to allocate the prizes when there are 20 athletes.
Answer for Question 1: 1,860,480
Question 2: Now, when there are only 10 total athletes, we perform the same calculation:
\[ P(10, 5) = \frac{10!}{(10 - 5)!} = \frac{10!}{5!} = 10 \times 9 \times 8 \times 7 \times 6 \]
Calculating that step-by-step:
- \( 10 \times 9 = 90 \)
- \( 90 \times 8 = 720 \)
- \( 720 \times 7 = 5040 \)
- \( 5040 \times 6 = 30240 \)
So, there are 30,240 ways to allocate the prizes when there are 10 athletes.
Answer for Question 2: 30240