To determine the number of ways 8 runners can finish the race, we need to consider all the possible permutations of the 8 runners. When we talk about permutations, we are referring to the different ways in which a set of items can be arranged.
Since we have 8 runners, we need to find the number of permutations of 8 distinct items. This is mathematically represented as [tex]\(8!\)[/tex] (8 factorial). The factorial of a number [tex]\(n\)[/tex] is the product of all positive integers less than or equal to [tex]\(n\)[/tex]. So, to find [tex]\(8!\)[/tex], we multiply all the integers from 1 to 8 together:
[tex]\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
Calculating this step by step:
1. [tex]\(8 \times 7 = 56\)[/tex]
2. [tex]\(56 \times 6 = 336\)[/tex]
3. [tex]\(336 \times 5 = 1680\)[/tex]
4. [tex]\(1680 \times 4 = 6720\)[/tex]
5. [tex]\(6720 \times 3 = 20160\)[/tex]
6. [tex]\(20160 \times 2 = 40320\)[/tex]
7. [tex]\(40320 \times 1 = 40320\)[/tex]
Therefore, the number of ways the 8 runners can finish the race is:
[tex]\[ 8! = 40320 \][/tex]
Given the choices available, the correct answer is [tex]\(8! = 40,320\)[/tex].
