There are five runners in a race. How many different ways can they cross the finish line?

[tex]
\begin{array}{c}
{ }_n P_r=\frac{n!}{(n-r)!} \
0!=1
\end{array}
[/tex]

Recognize that the order of runners matters, indicating a permutation problem.
Calculate the number of permutations using the factorial: 5 ! .
Compute 5 ! = 5 × 4 × 3 × 2 × 1 = 120 .
The number of different ways the runners can cross the finish line is 120 ​ .

Explanation

Understand the problem We have five runners in a race, and we want to determine the number of different ways they can cross the finish line. This is a permutation problem because the order in which the runners finish matters.

Recognize permutation Since the order matters, we need to calculate the number of permutations of the five runners. The number of permutations of n distinct objects is given by n ! , which is the factorial of n . In this case, n = 5 .

Calculate 5!
The factorial of 5, denoted as 5 ! , is calculated as follows: 5 ! = 5 × 4 × 3 × 2 × 1

Compute the result Calculating the product: 5 ! = 5 × 4 × 3 × 2 × 1 = 120 Thus, there are 120 different ways the five runners can cross the finish line.

State the final answer Therefore, the number of different ways the five runners can cross the finish line is 120.


Examples
In a school race with 5 participants, determining the number of possible finishing orders helps in planning awards and recognizing all potential outcomes. This concept extends to any competition where order matters, such as ranking teams or individuals based on performance.

Answered by GinnyAnswer | 2025-07-05

There are 120 different ways the five runners can cross the finish line, calculated using the permutation formula. This is determined by computing the factorial of 5, which is 5! = 120. Understanding permutations allows us to recognize that the order of finishing matters in a race.
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Answered by Anonymous | 2025-07-29