In how many ways can a coach form a football team from $20$ players?

Question

The question asks this:

1. A coach must form a football team from a group of $20$ candidates. How many structured teams (each player has a specific place: striker, goalkeeper, etc.) can he train?
2. How many unstructured teams (i.e., we only consider the group of $11$ selected players without worrying about who will be the center forward, the goalkeeper, etc.)?

I try in this way:

1) $C(20,4)= \dfrac{20!}{4!16!} = 4845$

2) $C(20,11)= \dfrac{20!}{11!9!} = 167960$

Is it correct? thank you in advance!

Answer 1

For the first part of your question:

There are $20$ people to choose from for $position1$ (let's say goalkeeper).

Now there are $19$ people to chose from for $position2$ (let's say striker).

...

and lastly there are $10$ people to chose from for $position11$ (let's say left wing back)

So you get $20*19*18*...*10$.

For the second part:

Your answer is correct. It is indeed $C(20,11)$. If we don't want the team to have any structure, we just choose $11$ from the $20$ people we have.