posted by Bryan .
Assume that the set S has 7 elements.
How many subsets of S have at most 3 elements
The total number of subsets = 2^7 = 128
This includes the null set
So what we want is
the null set, the set of singles, the set of doubles and the set of triples
= 1 + C(7,1) + C(7,2) + c(7,3)
= 1 + 7 + 21 + 35
So the number of subsets with at most 3 elements is 64