Question

Let $S=\{1,2,...,20\}$. A subset $B$ 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 :

• A. $\frac{5}{2^{20}}$
• B. $\frac{6}{2^{20}}$
• C. $\frac{4}{2^{20}}$
• D. $\frac{7}{2^{20}}$

Answer 1

The correct option is A $\frac{5}{2^{20}}$

$S=\{1,2,\cdots ,20\}$
Total number of subsets of $S$ is $2^{20}$.
Sum of elements of $S$ is $\frac{20\times 21}{2}=210$.
Sum of the elements of $B$ is $203$

So, we have to remove elements $x_1,x_2,...,x_n$ from $S$ such that $x_1+x_2+...+x_n=7$

The possible sets are
i.e. $\left\{7\right\},\{1,6\},\{2,5\},\{3,4\},\{1,2,4\}$
$\therefore$ Required probability $=\frac{5}{2^{20}}$