To determine the number of ways to allocate prizes in a race, we need to calculate the number of permutations of the top 5 positions among the total number of entrants.
Question 1: 20 total athletes
We need to find the number of permutations of 5 places from 20 athletes. This can be calculated using the formula for permutations:
\[ P(n, r) = \frac{n!}{(n-r)!} \]
Where \( n \) is the total number of items to choose from (in this case, athletes), and \( r \) is the number of items to choose (the top 5 positions).
Here, \( n = 20 \) and \( r = 5 \):
\[ P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!} = 20 \times 19 \times 18 \times 17 \times 16 \]
Calculating this step by step:
- \( 20 \times 19 = 380 \)
- \( 380 \times 18 = 6840 \)
- \( 6840 \times 17 = 116280 \)
- \( 116280 \times 16 = 1860480 \)
Therefore, the number of ways to allocate the prizes if 20 athletes enter is 1,860,480.
Question 2: 10 total athletes
Again, we use the permutation formula:
\[ P(10, 5) = \frac{10!}{(10-5)!} = \frac{10!}{5!} = 10 \times 9 \times 8 \times 7 \times 6 \]
Calculating this step by step:
- \( 10 \times 9 = 90 \)
- \( 90 \times 8 = 720 \)
- \( 720 \times 7 = 5040 \)
- \( 5040 \times 6 = 30240 \)
Therefore, the number of ways to allocate the prizes if 10 athletes enter is 30240.
Answers
- 1,860,480
- 30240