Assume that the set s has 8 elements. how many subsets of s have at most 2 elements?

C (n, k) = n! / k! (n - k) ! = number of k-combinations from a set of n elements.

The subsets of S that have 0, 1 or 2 elements include C (11,0) empty set, (C11,1) 1-element sets and (C11,2) 2-element sets.

C (11,0) = 11! / 0! 11! = 11! / 1*11! = 1

C (11,1) = 11! / 1! 10! = 10!*11 / 1*10! = 11

C (11,2) = 11! / 2! 9! = 9!*10*11 / 2*9! = 10*11/2 = 55

The total number of subsets of S that have at most 2 elements is

1 + 11 + 55 = 67