I'm doing a problem involving combinations as follows:
Consider a house with $n$ room-mates. Each weekend, one or more of the room-mates form a group to drive to a party. Of the group, one is the designated driver. How many ways can this be done? Do the calculation two ways:
The two ways are
where you pick the designated driver, then the rest of the room-mates going, and
where you pick everyone going, and you pick the designated driver from that group.
I am confused on how to calculate the left side. (I'm trying to show that they're equal.)
Thanks for helping!
Edit: I've come up with $C(n,1) \cdot C(n-1,k-1)$ for the left hand side, could be wrong though!