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.

Question

Question 1
If 20 total athletes enter the race, how many ways can the prizes be allocated?(1 point)
Responses

1,860,480
1,860,480

6
6

5,587,021,440
5,587,021,440

15,504
15,504
Question 2
If only 10 people enter the race, how many ways can the prizes be allocated?(1 point)
ways (type as a number only, no commas or symbols)

Answer 1

To determine the number of ways to allocate prizes to the top 5 finishers in a race, we can use permutations since the order in which the prizes are awarded matters.

Question 1

If there are 20 athletes and we need to choose the top 5, we can calculate the number of ways to do this using the permutation formula:

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

where \(n\) is the total number of athletes, and \(r\) is the number of positions to fill. In this case:

• \(n = 20\)
• \(r = 5\)

Calculating \(P(20, 5)\):

\[ P(20, 5) = \frac{20!}{(20-5)!} = \frac{20!}{15!} = 20 \times 19 \times 18 \times 17 \times 16 \]

Calculating the above step-by-step:

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

So, the number of ways to allocate the prizes to the top 5 finishers when there are 20 athletes is 1,860,480.

Question 2

If there are only 10 athletes, we again use the permutation formula:

\[ P(10, 5) = \frac{10!}{(10-5)!} = \frac{10!}{5!} = 10 \times 9 \times 8 \times 7 \times 6 \]

Calculating step-by-step:

1. \(10 \times 9 = 90\)
2. \(90 \times 8 = 720\)
3. \(720 \times 7 = 5040\)
4. \(5040 \times 6 = 30240\)

So, the number of ways to allocate the prizes to the top 5 finishers when there are 10 athletes is 30240.

Summary of Answers

1. 1,860,480
2. 30240