JEE Main 2014 — Mathematics Probability & Statistics

Let $\mathrm{S}=\left\{x_1, x_2, x_3, x_4, x_5, x_6, x_7\right\}$S={x1​,x2​,x3​,x4​,x5​,x6​,x7​} Let the chosen element be $\mathrm{x}_{\mathrm{i}}$xi​. Total number of subsets of $\mathrm{S}=2^7=128$S=27=128 No. of non-empty subsets of S $=128-1$=128−1 $=127$=127 We need to find number of those subsets that contains $x_i$xi​. \begin{array}{|l|l|l|l|l|l|l|} \hline 2 & 2 & 2 & 2 & 1 & 2 & 2 \ \hline \end{array} $x_1 x_2$x1​x2​-—- $x_i=-x_7$xi​=−x7​ For those subsets containing $x_i$xi​, each element has 2 choices. i.e., (included or not included) in subset, However as the subset must contain $x_i$xi​, $x_i$xi​ has only one choice. (included one) So, total no. of subsets containing $x_i=2 \times 2 \times 2 \times 2 \times 1 \times 2 \times 2=64$xi​=2×2×2×2×1×2×2=64 Required prob $$=\frac{\text { No. of subsets containing } x_i}{\text { Total no. of non-empty subsets }}$$= Total no. of non-empty subsets  No. of subsets containing xi​​ $$=\frac{64}{127}$$=12764​