Find the probability that two randomly selected subsets of 1,2,3,4,5 have exactly two elem…

Question

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

Accepted Answer

Choose the 2 common elements: ${}^5C_2$ ways. Each of the remaining 3 elements can independently be in $A$ only, $B$ only, or neither: $3^3$ ways.

$$P = \frac{{}^5C_2\times3^3}{2^5\times2^5} = \frac{10\times27}{2^{10}}=\frac{135}{2^9}$$

Solutions · 1