Question:

$A$ is a set containing elements. $P$ and $Q$ are two subsets of $A$. Then the number of ways of choosing $P$ and $Q$ such that $P \cap Q = \emptyset$ is

Updated On: Jan 29, 2026

2 2n-2n Cn
2n
3n-1
3n

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The Correct Option is D

Solution and Explanation

We are asked to find the number of ways to choose two subsets $ P $ and $ Q $ from a set $ A $ of $ n $ elements, such that $ P \cap Q = \emptyset $, meaning that $ P $ and $ Q $ are disjoint.

Step 1: Understanding the choices for each element of $ A $
Each element of $ A $ has three possibilities: 1. It can be in $ P $. 2. It can be in $ Q $. 3. It can be in neither $ P $ nor $ Q $. Since $ P $ and $ Q $ are disjoint, no element can be in both $ P $ and $ Q $ simultaneously.

Step 2: Counting the total number of ways
For each of the $ n $ elements of $ A $, there are 3 choices (either in $ P $, in $ Q $, or in neither). Therefore, the total number of ways to choose $ P $ and $ Q $ is: \[ 3^n \]

Step 3: Conclusion
Thus, the number of ways to choose $ P $ and $ Q $ such that $ P \cap Q = \emptyset $ is $ 3^n $.