Definitions

Definition permutation (order is important):

\(\displaystyle{P}{\left({n},{r}\right)}={\frac{{{n}!}}{{{\left({n}-{r}\right)}!}}}\)

Definition combination (order is not important):

\(C(n,r)=\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{n!}{r!(n-r)!}\)

with \(n!=n \cdot (n-1) \cdot \ldots \cdot 2 \cdot 1\)

Solution

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:

\(\displaystyle{P}{\left({5},{5}\right)}={\frac{{{5}!}}{{{\left({5}-{5}\right)}!}}}={\frac{{{5}!}}{{{0}!}}}={5}\ne{120}\)

Results:

120