Here’s a structured analysis and solution for a CAT 2023 Quantitative (Slot 1) problem, assuming it involves a Game Theory/Logic Puzzle (common in CAT exams). Below is a hypothetical example and step-by-step solution based on typical patterns:
Problem Statement
Example:
"Four friends—A, B, C, and D—must be assigned to four different tasks: Coding, Designing, Marketing, and Testing.
A cannot do Coding.
B cannot do Marketing.
C and D cannot be assigned to the same task.
How many valid assignments are possible?"
Solution
Step 1: Understand Constraints
A ≠ Coding
B ≠ Marketing
C and D ≠ Same Task
Step 2: Total Assignments Without Constraints
Total permutations = 4! = 24.
Step 3: Apply Constraints
Use Inclusion-Exclusion Principle or Case Analysis.
Case Analysis is often clearer for complex constraints.
Case 1: Assign A First
A can choose 3 tasks (Designing, Marketing, Testing).
Case 1.1: A chooses Designing
Remaining tasks: Coding, Marketing, Testing.
Assign B:
B cannot do Marketing → B has 2 options (Coding, Testing).
Subcase 1.1.1: B chooses Coding
Remaining tasks: Marketing, Testing.
Assign C and D:
C and D must take different tasks.
Valid permutations = 2! = 2.
Subcase 1.1.2: B chooses Testing
Remaining tasks: Coding, Marketing.
Assign C and D: 2! = 2.
Total for Case 1.1 = (2 + 2) = 4.
Case 1.2: A chooses Marketing
Remaining tasks: Coding, Designing, Testing.
Assign B:
B cannot do Marketing (already assigned to A) → B has 3 options.
Subcase 1.2.1: B chooses Coding
Remaining tasks: Designing, Testing.
Assign C and D: 2! = 2.
Subcase 1.2.2: B chooses Designing
Remaining tasks: Coding, Testing.
Assign C and D: 2! = 2.
Subcase 1.2.3: B chooses Testing
Remaining tasks: Coding, Designing.
Assign C and D: 2! = 2.
Total for Case 1.2 = 2 + 2 + 2 = 6.
Case 1.3: A chooses Testing
Remaining tasks: Coding, Designing, Marketing.
Assign B:
B cannot do Marketing → B has 2 options (Coding, Designing).
Subcase 1.3.1: B chooses Coding
Remaining tasks: Designing, Marketing.
Assign C and D: 2! = 2.
Subcase 1.3.2: B chooses Designing
Remaining tasks: Coding, Marketing.
Assign C and D: 2! = 2.
Total for Case 1.3 = 2 + 2 = 4.
Total Valid Assignments = 4 (Case 1.1) + 6 (Case 1.2) + 4 (Case 1.3) = 14.
Key Takeaways
Break Down Constraints: List all restrictions explicitly.
Use Case Analysis: Simplify problems by assigning variables step-by-step.
Check Overlaps: Ensure no two constraints conflict in subcases.
Time Management: Such problems require 2–3 minutes if approached methodically.
Common CAT Game Theory Topics:
Permutations/Combinations with restrictions.
Logic puzzles (seating arrangements, task assignments).
Graph theory (network flows, matchings).
Let me know if you need further clarification or additional examples!
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