Before You Begin

Question #40602

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$ . Two subsets $A$ and $B$ of $S$ , each containing exactly 4 elements, are chosen independently and uniformly at random. Let $E$ be the event that $1 \in A \cap B$ or $2 \in A \cap B$ . Given that $A \cap B \neq \emptyset$ , the probability of event $E$ occurring is $\frac{m}{n}$ , where $m$ and $n$ are coprime positive integers. Find the value of $m+n$ .

Options

A.
1411
B.
797
C.
285
D.
104

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Restricted Selection (At least one common)|Conditional Probability Constraints|Mutually Exclusive vs Independent

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle restricted selection within conditional probability spaces.

The primary trap lies in calculating the probability of the union of two events ( $1 \in A \cap B$ and $2 \in A \cap B$ ).

Students often incorrectly assume these events are independent (calculating the intersection probability as $1/16$ instead of $9/196$ ) or mutually exclusive (ignoring the intersection entirely).

Furthermore, the problem requires adjusting the sample space to account for the condition $A \cap B \neq \emptyset$ , penalizing those who forget to apply the conditional probability formula and simply calculate the unconditional probability.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Restricted Selection (At least one common)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free