Let A is a finite set contains n elements and 0 ≤ r ≤ n

Now take those subsets of A that have r elements each.

Now, Number of subsets of A having e elements each = ⁿC_r

Hence, total number of subsets of A = ⁿC₀ + ⁿC₁ + ⁿC₂ +........ ⁿC_n 

                                                 = (1 + 1)ⁿ    {from Binomial Theorem}

                                                 = 2ⁿ