Before You Begin

Question #40292

Difficulty: 8/10 | 🟢 Free

The Question

A subset of 6 distinct integers is chosen uniformly at random from the set $S = \{1, 2, 3, \dots, 20\}$ . Let $A$ be the event that the chosen subset contains at least one pair of consecutive integers. Let $B$ be the event that the difference between the largest and the smallest integers in the chosen subset is strictly greater than 9. What is the probability that neither event $A$ nor event $B$ occurs?

Options

A.
0
B.
$\frac{7}{1292}$
C.
$\frac{49}{1292}$
D.
$\frac{1001}{7752}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

The Empty Set Trap|De Morgan’s Laws in Probability|Gap Method (Non-adjacency)|Sample Space Construction

Why This Question Is Tricky

Common Mistakes

This question deploys the 'Empty Set Trap' disguised by De Morgan's Laws and the Gap Method.

Students often attempt to calculate $P(A^c)$ and $P(B^c)$ separately and combine them using assumed independence or inclusion-exclusion, missing that the intersection of the constraints is physically impossible.

Selecting 6 non-consecutive integers requires a minimum range of $(6-1) \times 2 = 10$ .

By restricting the range to $\le 9$ via the complement of event B, the favorable sample space becomes an empty set, making the probability strictly zero.

This tests mathematical intuition and existence verification over blind formula application.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	The Empty Set Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free