To determine the number of ways the prizes can be allocated to the top 5 finishers in a race with 20 participants, we can use the concept of permutations. We need to select 5 unique participants out of the 20 and arrange them in order, as their finish positions (1st, 2nd, 3rd, 4th, and 5th) matter.
The number of ways to arrange \( k \) participants out of \( n \) is given by the permutation formula:
\[ P(n, k) = \frac{n!}{(n-k)!} \]
In this case, \( n = 20 \) (the total number of participants) and \( k = 5 \) (the number of winners that will receive prizes).
Calculating \( P(20, 5) \):
\[ P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!} \]
This simplifies to:
\[ P(20, 5) = 20 \times 19 \times 18 \times 17 \times 16 \]
Now we calculate that:
-
Calculate \( 20 \times 19 \): \[ 20 \times 19 = 380 \]
-
Then multiply by 18: \[ 380 \times 18 = 6840 \]
-
Then multiply by 17: \[ 6840 \times 17 = 116280 \]
-
Finally, multiply by 16: \[ 116280 \times 16 = 1860480 \]
Thus, the total number of ways the prizes can be allocated to the top 5 finishers is:
\[ \boxed{1860480} \]