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Question #41004

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{1, 2, 3, 4, 5, 6, 7, 8\}$ and $A = (1, 6) \cap \mathbb{Z}$ . Find the total number of bijective functions $f: S \to S$ satisfying the following two conditions:

$f(x) \neq x$ for all $x \in S$ .
The image set $f(A)$ is closed under the binary operation $a \oplus b = |a - b| + 1$ .

Options

A.
121
B.
162
C.
317
D.
600

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Sample Space Construction|Derangements (Inclusion-Exclusion)|Closed vs Open Intervals|Binary Operation Closure

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to translate a binary closure condition into a specific set structure (Arithmetic Progressions starting at 1) and apply Inclusion-Exclusion for partial derangements.

The primary trap is twofold: students often miss the second valid subset {1, 3, 5, 7} (leading to the trap answer 121) and incorrectly apply the standard derangement formula D_4 instead of using PIE for partial derangements where the domain and codomain are not identical.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Sample Space Construction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free