Question

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

Answer 1

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