# 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 ]