Question

Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y,Z)$ that can be formed such that $Y\subseteq X,Z\subseteq X$ and $Y\cap Z$ is empty, is

• A. $5^3$
• B. $5^2$
• C. $3^5$
• D. $2^5$

Answer 1

The correct option is C $3^5$

$Y\cap Z=\varphi \Rightarrow Y$ and $Z$ are disjoint subsets of $X$.
Let $a\in X$
Case $1:a\in Y$ and $a\notin Z$
Case $2:a\in Z$ and $a\notin Y$
Case $3:a\notin Y$ and $a\notin Z$
So, each element has $3$ options.
There are $5$ possibilities for $a$.
$\therefore$ Possible number of ordered pairs $=3^5$