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Question #41138

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all sequences $(x_1, x_2, x_3)$ of positive integers such that $x_1 + x_2 + x_3 = 92$ . A function $f: S \to \mathbb{Z}$ is defined by $f(x_1, x_2, x_3) = x_1 + 2x_2 + 3x_3$ . If $A \subseteq S$ is the largest possible subset such that the restriction $f|_A$ is injective, find the cardinality of $A$ .

Options

A.
179
B.
4095
C.
181
D.
88

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

Composite Injectivity/Surjectivity|Insertion of Means|Multinomial Theorem|Complementary Counting

Why This Question Is Tricky

Common Mistakes

This question tests the concept of composite injectivity and complementary counting within a multinomial domain.

The trap lies in confusing the domain size with the range size.

Students often calculate the total number of valid sequences using the Multinomial Theorem (which yields 4095) and assume the function is injective over the entire domain, failing to account for the linear dependence of the function's coefficients with the sum constraint.

By requiring the maximum subset for injectivity, the problem forces students to find the exact cardinality of the image of the function, filtering out those who over-count.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Composite Injectivity/Surjectivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free