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 $3^5$

$Y\subseteq X,Z\subseteq X$

Let $a\in X$, then

$a\in Y,a\in Z.....\left(1\right)$

$a\notin Y,a\in Z.....\left(2\right)$

$a\in Y,a\notin Z.....\left(3\right)$

$a\notin Y,a\notin Z.....\left(4\right)$

We require $Y\cap Z=\phi$

Thus, for $a$, equation $\left(1\right),\left(2\right)$ and $\left(3\right)$ can satisfy $Y\cap Z=\phi$

Therefore,

$Y\cap Z=\phi$ has $3$ options.

Thus,

No. of required possibilities for $5$ elements of $X=3^5$