Let S = {1, 2, ... , 20). A and B subset of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a randomly chosen subset of S is "nice" is

$S=\{1,2,\dots ,20\}$

Total no. of subsets =$2^{20}$

We have, $1+2+\dots +20=\frac{20\left(21\right)}{2}=210$

Sum of elements will be 203 if we would not choose 7 or elements which are giving sum 7.

Elements giving sum 7 are (1, 6), (2, 5), (3, 4), (1, 2, 4) Thus, a subset will be 'nice' if we didn't choose these 5 outcomes.

$\therefore$  Required probability (subset choosen is 'nice') $=\frac{5}{2^{20}}$