The number of subsets of the set A=a1,a2,…,an which contain even number of elements is

For each of the first (n – 1) elements  a1, a2 ….…,an−1 we have two choices: either ai(1≤i≤n−1) lies in the subset or ai doesn’t lie in the subset. For the last element  we have just one choice. If even number of elements have  already been selected, we do not include an in the subset  otherwise (when odd number of elements have been selected) we include it in the subset Thus, the number of subsets of A=a1,a2,…,an which contain even number of elements is equal to 2n−1