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

Answer: (B)

Here, $X=\{1,2,3,4,5\}$
Now, the possibilities are
(i) $1 \in Y, 1 \in Z$
(ii) $1 \in Y, 1 \notin Z$
(iii) $1 \notin Y, 1 \in Z$
(iv) $1 \notin Y, 1 \notin Z$
(ii) (iii) ,(iv) are favourable cases for $Y \cap Z$ to be empty.
Similarly , for other elements we have 3 favourable cases.
$\therefore$ Required number of different ordered pairs $(Y, Z)=3 \times 3 \times 3 \times 3 \times 3=3^5$