The probability that two randomly selected subsets of the set

Question:

The probability that two randomly selected subsets of the set $\{1,2,3,4,5\}$ have exactly two elements in their intersection, is :

 

  1. $\frac{65}{2^{7}}$

  2. $\frac{65}{2^{8}}$

  3. $\frac{135}{2^{9}}$

  4. $\frac{35}{2^{7}}$


Correct Option: , 3

Solution:

Total subsets $=2^{5}=32$

Probability $=\frac{{ }^{5} \mathrm{C}_{2} \times 3^{3}}{32 \times 32}=\frac{10 \times 27}{12^{10}}=\frac{135}{2^{9}}$

 

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