A set S contains 7 elements. A non-empty subs | vedclass

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Question

Let $\mathrm{S}=\left\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}\right\}$

Let the chosen element be $x_{i}$,

Total number of subsets of $S

Vedclass · Accepted Answer

$\frac{64}{127}$

Vedclass · Answer

Let $\mathrm{S}=\left\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}ight\}$

Let the chosen element be $x_{i}$,

Total number of subsets of $S=2^{7}=128$

No. of non-empty subsets of $S=128-1$ $=127$

We need to find number of those subsets that contains $x_{i}$,

$\boxed2\boxed2\boxed2\boxed2\boxed1\boxed2\boxed2$

${x_1}\,{x_2} -  - {x_i} -  - {x_7}$

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 $x_{i}$ $x_{f}$ has only one choice. (included one)

So, total no. of subsets containing $x_{i}=2 	imes 2 	imes 2 	imes 2 	imes 1 	imes 2 	imes 2=64$

Required prob

$ = \frac{{No.\,\,of\,\,subsets\,\,containing\,\,{x_i}}}{{Total\,\,no.\,\,of\,non - empty\,\,subsets}}$

$=\frac{64}{127}$

A set $S$ contains $7$ elements. A non-empty subset $A$ of $S$ and an element $x$ of $S$ are chosen at random. Then the probability that $x \in A$ is

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