# 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?

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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The problem reads: Suppose $$\displaystyle{P}{\left({X}_{{1}}\right)}={.75}$$ and $$\displaystyle{P}{\left({Y}_{{2}}{\mid}{X}_{{1}}\right)}={.40}$$. What is the joint probability of $$\displaystyle{X}_{{1}}$$ and $$\displaystyle{Y}_{{2}}$$?
This is how I answered it. P($$\displaystyle{X}_{{1}}$$ and $$\displaystyle{Y}_{{2}}$$) $$\displaystyle={P}{\left({X}_{{1}}\right)}\times{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}={.75}\times{.40}={0.3}.$$
What I don't understand is how do you get the $$\displaystyle{P}{\left({Y}_{{1}}{\mid}{X}_{{1}}\right)}$$? I am totally new to Statistices and I need to understand each part of the process in order to get the whole concept. Can anyone help me to understand why the P and X exist and what they represent?