Q119 / 180 · COMEDK 2017 · Mathematics

The total number of subsets of a finite set $A$ has 56 more elements than the total number of subsets of another finite set $B$. What is the number of elements in the set A?

  • A
    5
  • Correct answer
    B
    6
  • C
    7
  • D
    8

Solution

Let sets $A$ and $B$ have $m$ and $n$ elements, respectively. Then, $2^{m}-2^{n}=56 \Rightarrow 2^{n}\left(2^{m-n}-1\right)=56$ $\begin{array}{ll} \Rightarrow & 2^{n}\left(2^{m-n}-1\right)=8 \times 7 \\ \Rightarrow & 2^{n}\left(2^{m-n}-1\right)=2^{3} \times 7 \end{array}$ One comparing both sides, we get $\begin{array}{ll} & 2^{n}=2^{3} \text { and } 2^{m-n}-1=7 \\ \Rightarrow & n=3 \text { and } 2^{m-n}=8 \\ \Rightarrow & 2^{m-n}=2^{3} \\ \Rightarrow & m-n=3 \\ \Rightarrow & m-3=3 \\ \Rightarrow & m=6 \end{array}$ Hence, number of the elements is set $A$ is 6 .

Hence, the correct answer is option: 'B'.

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