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

Answer 1

The correct option is B

$3^5$

Explanation for the correct option:

Find the number of different ordered pairs to satisfy the given condition:

Given, $Y\subseteq X,Z\subseteq X$

$X=\{1,2,3,4,5\}$

Let us choose an element $a\in X$, then

$a\in Y\text{and}a\in Z$ $\to \left(1\right)$

$a\in Y\text{and}a\notin Z$ $\to \left(2\right)$

$a\notin Y\text{and}a\in Z$ $\to \left(3\right)$

$a\notin Y\text{and}a\notin Z$ $\to \left(4\right)$

Given, condition $Y\cap Z=\varphi$ is satisfied by equation $2,3,4$.

Therefore,$Y\cap Z=\varphi$ has three possibilities.

The required number of possibilities for one element is $3$.

The required number of possibilities for five element is $3\times 3\times 3\times 3\times 3$. $=3^5$

Hence, option (B) is the correct answer.