A sports team has 7 players. In how many ways can the team select a captain and junior captain?

Question
A sports team has 7 players. In how many ways can the team select a captain and junior captain?

Answers (1)

2020-12-06
Definition permutation (order is important): \(\displaystyle{P}{\left({n},{r}\right)}={n}\frac{!}{{{n}-{r}}}!\)
Definition combination (order is not important):
\(\displaystyle{C}{\left({n},{r}\right)}={\left(\frac{{n}}{{r}}\right)}={n}\frac{!}{{{r}!{\left({n}-{r}\right)}!}}\)
with \(\displaystyle{n}\ne{n}\cdot{\left({n}-{1}\right)}\cdot\ldots\cdot{2}\cdot{1}.\)
We are interested in selecting two of the seven players.
n=7
r=2
The order of the players is important (as a diffirent order results in a diffirent captain and junior captain), thus we need to use the definition or permutation.
Evaluate the definition of permutation at n=7 and r=2:
\(\displaystyle{7}{P}{2}={7}\frac{!}{{{7}-{2}}}\ne{7}\frac{!}{{5}}\ne\frac{{{7}\cdot{6}\cdot{5}!}}{{5}}\ne{7}\cdot{6}={42}\)
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