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:

\(\displaystyle{7}{P}{2}={7}\frac{!}{{{7}-{2}}}\ne{7}\frac{!}{{5}}\ne\frac{{{7}\cdot{6}\cdot{5}!}}{{5}}\ne{7}\cdot{6}={42}\)

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:

\(\displaystyle{7}{P}{2}={7}\frac{!}{{{7}-{2}}}\ne{7}\frac{!}{{5}}\ne\frac{{{7}\cdot{6}\cdot{5}!}}{{5}}\ne{7}\cdot{6}={42}\)