# In how many different orders can five runners finish a race if no ties are allow

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

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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