# A sports team has 7 players. In how many ways can the team select a captain and junior captain? Question
Probability and combinatorics 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:
$$\displaystyle{7}{P}{2}={7}\frac{!}{{{7}-{2}}}\ne{7}\frac{!}{{5}}\ne\frac{{{7}\cdot{6}\cdot{5}!}}{{5}}\ne{7}\cdot{6}={42}$$

### Relevant Questions 9 students are in a math class. How many different ways can you choose 6 people for a group? Suppose a class consists of 5 students majoring in Computer Science, 5 students majoring in Chemistry and 3 students majoring in Mathematics. How many ways are possible to form a group of 3 students if each group should consist at most 2 students majoring in Computer Science? If Jeremy has 4 times as many dimes as nickels and they have a combined value of 360 cents, how many of each coin does he have? Suppose a class consists of 4 students majoring in Mathematics, 3 students majoring in Chemistry and 4 students majoring in Computer Science. How many compositions are possible to form a group of 3 students if each group should consist at most 2 students majoring in Computer Science? In one study, the correlation between the educational level of husbands and wives in a certain town was about 0.50, both averaged 12 years of schooling completed, with an SD of 3 years.
a) Predict the educational level of a woman whose husband has completed 18 years of schooling b) Predict the educational level of a man whose wife has completed 15 years of schooling. c) Apparently, well-educated men marry women who are less well educated than themselves. But the women marry men with even less education. How is this possible? 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?    