JEE Main 2014 — Probability Question with Solution

Let $\mathrm{S}=\left\{x_1,x_2,x_3,x_4,x_5,x_6,x_7\right\}$ Let the chosen element be ${\mathrm{x}}_{\mathrm{i}}$. Total number of subsets of $\mathrm{S}=2^7=128$ No. of non-empty subsets of S $=128-1$ $=127$ We need to find number of those subsets that contains $x_i$. \begin{array}{|l|l|l|l|l|l|l|} \hline 2 & 2 & 2 & 2 & 1 & 2 & 2 \\ \hline \end{array} $x_1{x}_2$-—- $x_i=-x_7$ For those subsets containing $x_i$, each element has 2 choices. i.e., (included or not included) in subset, However as the subset must contain $x_i$, $x_i$ has only one choice. (included one) So, total no. of subsets containing $x_i=2\times 2\times 2\times 2\times 1\times 2\times 2=64$ Required prob $$=\frac{\text{ No. of subsets containing }{x}_i}{\text{ Total no. of non-empty subsets }}$$ $$=\frac{64}{127}$$