Question

Parker is organizing a race. Only the top 5 will be awarded money as follows: 1st place = $1,000; 2nd place = $800; 3rd place = $600; 4th place = $400; and 5th place = $200.

If only 20 people enter the race, how many ways cn the prizes be allocated?

Answer 1

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:

1. Calculate \( 20 \times 19 \):
\[
20 \times 19 = 380
\]

2. Then multiply by 18:
\[
380 \times 18 = 6840
\]

3. Then multiply by 17:
\[
6840 \times 17 = 116280
\]

4. 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:

\[
\boxed{1860480}
\]