Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y…
Let $X=\{1,2,3,4,5\}$. The number of different ordered pairs $(Y, Z)$ that can be formed such that $Y \subseteq X, Z$ $\subseteq \mathrm{X}$ and $\mathrm{Y} \cap \mathrm{Z}$ is empty, is
$5^2$
$3^5$
$2^5$
$5^3$
Solution
$\mathrm{Y} \subseteq \mathrm{X}, \mathrm{Z} \subseteq \mathrm{X}$
Let $a \in X$, then we have following chances that
(1) $a \in Y, \quad a \in Z$
(2) $a \notin Y, \quad a \in Z$
(3) $a \in Y, \quad a \notin Z$
(4) $a \notin Y, \quad a \notin Z$
We require $Y \cap Z=\phi$
Hence (2), (3), (4) are chances for '$a$' to satisfy $Y \cap Z=\phi$.
$\therefore Y \cap Z=\phi$ has $3$ chances for $a$.
Hence for five elements of $\mathrm{X}$, the number of required chances is $3 \times 3 \times 3 \times 3 \times 3=3^5$