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Question #40313

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be the set of all possible values of $k$ such that a selection of exactly $k$ items can be made from a pool of $38$ identical objects. Let $B$ be the set of all possible values for the number of linear arrangements of $k$ distinct objects, where $k \in A$ . Find the cardinality of the symmetric difference, $|A \Delta B|$ .

Options

A.
68
B.
69
C.
70
D.
73

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

Difference vs. Symmetric Difference|Selection from Identical Items|Permutations of Identical Objects

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to correctly identify the sample space of selections from identical items versus arrangements of distinct items, specifically targeting the 'zero-selection' (null set) case.

Students frequently make an off-by-one error by assuming selections start from $k=1$ , leading to $A = \{1, \dots, 38\}$ and $|A|=38$ .

Consequently, they calculate $|B|=38$ and $|A \cap B|=4$ , yielding a symmetric difference of $38 + 38 - 8 = 68$ (the lethal trap).

Furthermore, the question tests the property of sets where duplicate elements are collapsed, requiring the realization that $0! = 1! = 1$ , which reduces the true cardinality of $B$ from 39 to 38.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Difference vs. Symmetric Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free