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Question #41022

Difficulty: 8/10 | 🟢 Free

The Question

Let $A = \{1, 2, 3, 4, 5, 6\}$ and $B = \left\{ x \in \mathbb{N} : x \le 5 \text{ and } \frac{x-3}{x^2-5x+4} \in \mathbb{R} \right\}$ . Find the number of surjective functions $f: A \to B$ such that $f(1) + f(2) = 7$ .

Options

A.
130
B.
144
C.
162
D.
260

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

Combinatorial Counting|Surjective Mapping|Bijectivity and Inverse|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the integration of combinatorial counting (Inclusion-Exclusion) with domain/codomain constraints.

The primary trap is the 'Denominator Restriction' acting on the codomain.

Students often gloss over the condition that the rational expression must be real, incorrectly assuming $B = \{1, 2, 3, 4, 5\}$ .

This false assumption inflates the number of valid pairs for $f(1) + f(2) = 7$ (adding $(3,4)$ and $(4,3)$ ) and fundamentally alters the surjective PIE calculation, leading to the trap answer of 144.

Recognizing the restricted codomain $B = \{2, 3, 5\}$ collapses the problem into a constrained partition where two elements are pre-covered, requiring a modified PIE approach.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Combinatorial Counting (P&C Integration)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free