Question

Let $X = \{a,\,b\}$ and $Y = \{1,\,3,\,4,\,5\}$. A subset of $X\times Y$ is selected at random. If $A$ is an event of selecting a subset of $X\times Y$ containing exactly three elements, then $P(A) =$

• A. $\dfrac{7}{32}$
• B. $\dfrac{7}{64}$
• C. $\dfrac{5}{32}$
• D. $\dfrac{5}{64}$
• E. $\dfrac{7}{128}$

Hint:

The Cartesian product $X\times Y$ has $|X|\cdot|Y|$ elements. When selecting a subset at random from a set of $n$ elements, the sample space has $2^n$ equally likely outcomes.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
$|X\times Y| = |X|\cdot|Y| = 2\times 4 = 8$. Total number of subsets of $X\times Y$ is $2^8 = 256$. Count the subsets with exactly 3 elements.

Step 2: Detailed Explanation:
Number of subsets of $X\times Y$ with exactly 3 elements $= \binom{8}{3} = 56$.
\[ P(A) = \frac{56}{256} = \frac{7}{32} \]

Step 3: Final Answer:
$P(A) = \dfrac{7}{32}$.