Question

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

Short answer

The resultant answer is 120 .

Step-by-step solution

Given data

The given data is five runners.

Concept of Permutation

Definition permutation (order is important):$\mathbf{P}\mathbf{(}\mathbf{n}\mathbf{,}\mathbf{r}\mathbf{)}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}$

Definition combination (order is not important):$\mathbf{C}\mathbf{(}\mathbf{n}\mathbf{,}\mathbf{r}\mathbf{)}\mathbf{=}\left(\begin{array}{l}n\\ r\end{array}\right)\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}$

with $\mathbf{n}\mathbf{!}\mathbf{=}\mathbf{n}\mathbf{\cdot }\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\cdot }\mathbf{\dots }\mathbf{\cdot }\mathbf{2}\mathbf{\cdot }\mathbf{1}$.

Evaluate the definition of a combination

The order of the runners is important; thus, we need to use the definition of permutation.

We will select 5 runners from the 5 runners (as we want an ordering of all runners).

n = 5

r = 5

Evaluate the definition of a combination:

$\begin{array}{r}P(5,5)=\frac{5!}{(5-5)!}\\ =\frac{5!}{0!}\\ =5!\\ =120\end{array}$