Why does 5 * (13 choose 5) represent: the number of ways you can choose a team of five people chosen from a room of 13 people, where you are designating the captain? I understand the choose part, but not the 5 multiplier.

Question

unklerhaukus · Answer

there can be 5 different captains

anonymous · Answer

I know that, but in the next "blank" why would it then be 13 choose 5? That isn't the number of people available remaining that we are allowed to pick.

anonymous · Answer

I really hope that makes sense.

anonymous · Answer

Shouldn't we only pick 4 after?

unklerhaukus · Answer

$$^{13}C_5$$ is the number of possible teams,

$$5\times~^{13}C_5$$ is the number of possible teams where the captain is also chosen

anonymous · Answer

Oh, so there are 6 people all together?

unklerhaukus · Answer

no the captain is one of the 5 member in the team

unklerhaukus · Answer

members*

anonymous · Answer

Are we not already picking 5 people with $$\left(\begin{matrix}13 \5 \end{matrix}\right)$$

unklerhaukus · Answer

yeah,

anonymous · Answer

So if we've chosen our 5, how would multiplying by 5 affect it?

anonymous · Answer

This is tricky.

unklerhaukus · Answer

im not sure what part you arn't getting @DanielHendrycks ?

anonymous · Answer

I now see. We are picking from the 5 people one captain. I thought we were picking from the entire pool a captain. There are 5 possibilities for captain and multiplying the two gives the numbers of varieties we can have of them (together).

unklerhaukus · Answer

thats right

unklerhaukus · Answer

$$^{13}C_5\times ^5C_1$$