Probability 1 Question 4
4. Let $S={1,2, \ldots, 20}$. $A$ 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) $\frac{6}{2^{20}}$
(b) $\frac{4}{2^{20}}$
(c) $\frac{7}{2^{20}}$
(d) $\frac{5}{2^{20}}$
(2019 Main, 11 Jan II)
Show Answer
Answer:
Correct Answer: 4. (d)
Solution:
- Number of subset of $S=2^{20}$
Sum of elements in $S$ is $1+2+\ldots .+20=\frac{20(21)}{2}=210$
$$ \because 1+2+\ldots \ldots+n=\frac{n(n+1)}{2} $$
Clearly, the sum of elements of a subset would be 203, if we consider it as follows
$S-{7}, S-{1,6} S-{2,5}, S-{3,4}$ $S-{1,2,4}$
$\therefore$ Number of favourables cases $=5$
Hence, required probability $=\frac{5}{2^{20}}$
