Ronin Daniels

2023-03-29

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

Karma Gutierrez

Beginner2023-03-30Added 2 answers

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:

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

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:

$P(5,5)=\frac{5!}{(5-5)!}$

We have 5 runners (n = 5) and we want to arrange all of them (r = 5).

Simplifying the expression, we have:

$P(5,5)=\frac{5!}{0!}$

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:

$P(5,5)=\frac{5!}{1}$

Calculating the factorial:

$5!=5\times 4\times 3\times 2\times 1=120$

Substituting the values back into the expression, we have:

$P(5,5)=\frac{120}{1}=120$

Therefore, the number of different orders in which five runners can finish the race with no ties allowed is 120.

So, $P(5,5)=120$.

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:

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

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:

$P(5,5)=\frac{5!}{(5-5)!}$

We have 5 runners (n = 5) and we want to arrange all of them (r = 5).

Simplifying the expression, we have:

$P(5,5)=\frac{5!}{0!}$

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:

$P(5,5)=\frac{5!}{1}$

Calculating the factorial:

$5!=5\times 4\times 3\times 2\times 1=120$

Substituting the values back into the expression, we have:

$P(5,5)=\frac{120}{1}=120$

Therefore, the number of different orders in which five runners can finish the race with no ties allowed is 120.

So, $P(5,5)=120$.

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