Let $\mathrm{S}=\{1,2, \ldots . ., 20\}$. A subset $\mathrm{B}$ of $\mathrm{S}$ is said to be "nice", if the…
Let $\mathrm{S}=\{1,2, \ldots . ., 20\}$. A subset $\mathrm{B}$ of $\mathrm{S}$ is said to be "nice", if the sum of the elements of $\mathrm{B}$ is 203 . Than the probability that a randomly chosen subset of $S$ is "nice" is :
$\frac{7}{2^{20}}$
$\frac{5}{2^{20}}$
$\frac{4}{2^{20}}$
None of the above
Solution
Since total number of subsets of the set $S=2^{20}$
Now, the sum of all number from 1 to $20=\frac{20 \times 21}{2}=210$ Then, find the sets which has sum 7 .
(1) $\{7\}$
(2) $\{1,6\}$
(3) $\{2,5\}$
(4) $\{3,4\}$
(5) $\{1,2,4\}$
Then, there is only 5 sets which has sum 203
Hence required probability $=\frac{5}{2^{20}}$