How many ordered pairs $(A,B)$ they are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5 \}$ such that $A \cap B = \{1\}$?
How many ordered pairs $(A,B)$ they are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5 \}$ such that $A \cap B = \{1\}$?
If 1 is in both sets :
Size of $A$ is $>1$ and size of $B$ is $>1$.
$\Rightarrow$ Size of $A \cup B =4$
$$5 \cdot (3 \cdot 3 \cdot 3 \cdot 3) = 405$$ is not correct.
What am I missing here?
Each of the elements $2, 3, 4, 5$ can be in $A, B$, or neither. These possibilities are mutually exclusive and exhaustive. So the number of possible ordered pairs of subsets that work is $3^4=81$.