Question

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

• A. $\frac{65}{2^7}$
• B. $\frac{135}{2^9}$
• C. $\frac{65}{2^8}$
• D. $\frac{35}{2^7}$

Answer 1

The correct option is B $\frac{135}{2^9}$

Let $A$ and $B$ be two subsets.
For each $x\in \{1,2,3,4,5\},$ there are four possibilities :
$x\in A\cap B,x\in A^{\prime }\cap B,x\in A\cap B^{\prime },x\in A^{\prime }\cap B^{\prime }$
So, the number of elements in sample space $=4^5$
Required probability
$=\frac{{}^5{C}_2\times 3^3}{4^5}$
$=\frac{10\times 27}{2^{10}}=\frac{135}{2^9}$