Prove that the total number of subsets of a finite set containing n elements is $2^n$.

Solution :

Let A be a finite set containing n elements. Let 0 $\le$ r $\le$ n.

Consider those subsets of A that have r elements each. We know that the number of ways in which r elements can be chosen out of n elements is $^nC_r$.

Therefore, the number of subsets of A having r elements each is $^nC_r$.

Hence, the total number of subsets of A

= $^nC_0$ + $^nC_1$ + $^nC_2$ + …. + $^nC_n$ = $(1 + 1)^n$ = $2^n$.

[ Using binomial theorem ]