In how many ways can a woman invite a group of one or more friends from a group of six friends out to dinner? I'm confident the answer is $2^{6}-1$, but I wanted to know if my reasoning is correct. We have $2^{6}$ since there are $2^{1}$ ways to invite a friend, and we have six of them, so we have $2*2*2*2*2*2$ ways to invite six friends, but we subtract by $1$ so that at least one friend is present, meaning we don't care which friend it is, but we get rid of the case of not inviting a friend for one of the six friends. Does this sound correct? 
 A: As you correctly point out, the reason why you subtract one, is because you are removing the case in which no friends are invited, from the total number of ways to invite the $6$ friends. 
The total number of ways to invite $6$ friends is $2^6$, since for each friend their are two options, 'invite' or 'do not invite'. But, you want to remove the case where the 'do not invite' option is chosen for all of the $6$ friends, and so the total number of ways to invite at least one friend is $2^6 - 1$.
An alternative solution is as follows. The number of ways to choose one or more friend is $$\sum_{k=1}^6 C(6,k).$$ She chooses one friend, or she chooses two friends, etc. Using the fact that $2^n = \sum_{k=0}^n C(n,k)$, we see that the number of ways she can invite one or more friend is $2^6 - C(6,0) = 2^6 - 1$.
A: That's right. The question is basically - how many non-empty subsets are there for a set of $6$ elements.
The size of a power set of a set with $6$ elements is $2^6$, but since your don't count the empty set, you get $2^6-1$.
