Question

4 fill in the blank 15 points there are 8 people running a race. how many different outcomes for the race are there? type your answer... there are 8 people competing in a race. in how many different ways can first, second, and third place medals be awarded? type your answer...

Explanation:

First Sub - Question (Number of different race outcomes)

Step1: Identify the problem type

This is a permutation problem where we want to find the number of ways to arrange 8 people (since each outcome is a different order of the 8 runners). The formula for permutations of \(n\) objects taken \(n\) at a time is \(n!\) (n factorial), where \(n!=n\times(n - 1)\times(n - 2)\times\cdots\times1\). Here, \(n = 8\).

Step2: Calculate \(8!\)

\(8! = 8\times7\times6\times5\times4\times3\times2\times1\)
\(8\times7 = 56\), \(56\times6=336\), \(336\times5 = 1680\), \(1680\times4=6720\), \(6720\times3 = 20160\), \(20160\times2=40320\), \(40320\times1 = 40320\)

Second Sub - Question (Number of ways to award 3 medals)

Step1: Identify the problem type

This is a permutation problem where we want to find the number of ways to arrange 8 people taken 3 at a time. The formula for permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 8\) and \(r=3\).

Step2: Calculate \(P(8,3)\)

First, find \((n - r)!\) where \(n = 8\) and \(r = 3\), so \((8-3)!=5! = 5\times4\times3\times2\times1=120\)
\(n!=8!=40320\) (from the first sub - question)
Then \(P(8,3)=\frac{8!}{(8 - 3)!}=\frac{40320}{120}=336\)

Answer:

For the first blank: \(40320\)
For the second blank: \(336\)