Number of ways to pick one or more elements from a set of $N$ elements I'm looking for a particular formula in combinatorics, but I don't even know what it would be called.   It is likely related to the Bell Number.
Given a set of $N$ elements, I'd like to know how many different ways I can pick one-or-more elements from the set.
For example: $N = 3$  (set consisting of a, b, and c)

*

*a

*b

*c

*a + b

*a + c

*b + c

*a + b + c
(The Bell Number for a set of size 3 is only 5, so it comes up short of what I'm looking for)
Is there a term for this value, and how can it be calculated?

If you want to select one or more elements from a set, you are selecting a non-empty subset. So you are looking for a way to count the number of non-empty subsets of a set. 
This is exactly the cardinality of the powerset of your set minus 1 (throw away the empty set). For a set of size $n$, its powerset has $2^n$ elements. Thus the number of ways of selecting one or more elements from a set of size $n$ is $2^n-1$.
You can also do this with combinations. Recall that ${n \choose k} = \frac{n!}{k!(n-k)!}$ is the number of ways of choosing $k$ objects from $n$ objects. Looking at it this way you have: $${n \choose 1} + {n \choose 2} + \cdots + {n \choose n} = -1 + {n \choose 0} + {n \choose 1} + \cdots + {n \choose n} = -1 + (1+1)^n = 2^n-1$$
where I used the binomial theorem to get the next to last equality.
