TITLE: Number of different ordered pairs
QUESTION [7 upvotes]: Let $X = \{1,2,3,4,5\}$. What is the number of different ordered pairs $(Y,Z)$ that can formed such that
$$Y\subseteq  X$$
$$Z\subseteq X$$
$$Y\cap Z=\emptyset$$
How to make cases in this question?

REPLY [3 votes]: Here's another way to think about the problem.
Each element can be 'colored' with the color $Y$, $Z$ or $0$, based on whether it belongs to $Y$, $Z$ or $0$. Since $Y \cap Z = \emptyset$, each element can be assigned exactly one color for every possible ordered pair. For example, $Y = \{ 1,2,3\}$ and $Z= \{4 \}$ would give the coloring $YYYZ0$.
There are 3 different colors and 5 symbols, so $3^5$ possible colorings. Each coloring corresponds to a pair $(Y,Z)$, so there are $3^5$ such ordered pairs.