Question

Of 20 athletes who will qualify for a race, only the top 5 will win prize money: 1st=$1,000; 2nd=$800; 3rd=$600; 4th=$400; and 5th=$200. How many ways can the top 5 awards be allocated from a field of 20?

Answer 1

To determine how many ways the top 5 awards can be allocated from a field of 20 athletes, we first recognize that this is a permutation problem, since the order in which athletes finish is important (i.e., 1st, 2nd, 3rd, 4th, and 5th place).

The number of ways to choose and arrange 5 athletes out of 20 can be calculated using the permutation formula:

\[
P(n, k) = \frac{n!}{(n-k)!}
\]

where:
- \( n \) is the total number of athletes (20),
- \( k \) is the number of top positions to fill (5).

Plugging in the values:

\[
P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!}
\]

To simplify \( \frac{20!}{15!} \):

\[
\frac{20!}{15!} = 20 \times 19 \times 18 \times 17 \times 16
\]

Now we will calculate this product step-by-step:

1. \( 20 \times 19 = 380 \)
2. \( 380 \times 18 = 6840 \)
3. \( 6840 \times 17 = 116280 \)
4. \( 116280 \times 16 = 1860480 \)

Therefore, the number of ways to allocate the top 5 awards to the 20 athletes is:

\[
\boxed{1860480}
\]