Question

How many subsets of a set with 100 elements have more than one element?

Answer 1

2^100 - 101 This is a fairly simple question. The first part is "How many possible subsets can you have of a set with 100 elements?" For that, consider that for each element in the original set, you can either have or not have that element in the subset. And since there's 100 elements in the original set, you can have 2^100 possible subsets ranging from the empty set, all the way to the subset that is the whole original set. Hence, the 2^100 in the answer. Now let's subtract from that the empty set, giving us 2^100 - 1. And then subtract all the subsets with a single member from the original set. There's 100 of those, so you have 2^100 - 1 - 100, which in turn simplifies to 2^100 - 101. Which in turn is: 903,380,628,834,704,070,774,685,861,275