Ronin Daniels

2023-03-29

In how many different orders can five runners finish a race if no ties are allowed???

Karma Gutierrez

To determine the number of different orders in which five runners can finish a race with no ties allowed, we can use the concept of permutations.
In a permutation, the order of arrangement matters. We want to find the number of permutations of the five runners.
The formula to calculate the number of permutations is given by:
$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$
where $n$ is the total number of objects (in this case, the number of runners) and $r$ is the number of objects to be arranged (in this case, also the number of runners).
Let's calculate the number of permutations for this problem:
$P\left(5,5\right)=\frac{5!}{\left(5-5\right)!}$
We have 5 runners (n = 5) and we want to arrange all of them (r = 5).
Simplifying the expression, we have:
$P\left(5,5\right)=\frac{5!}{0!}$
The factorial of a number is the product of all positive integers less than or equal to that number. By definition, $0!$ is equal to 1.
Therefore, we can rewrite the expression as:
$P\left(5,5\right)=\frac{5!}{1}$
Calculating the factorial:
$5!=5×4×3×2×1=120$
Substituting the values back into the expression, we have:
$P\left(5,5\right)=\frac{120}{1}=120$
Therefore, the number of different orders in which five runners can finish the race with no ties allowed is 120.
So, $P\left(5,5\right)=120$.

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