CAT 2024 Most Difficult Slot: Solving the Permutation-Combination Puzzler
The CAT 2024 exam posed a notoriously challenging question in the Permutation-Combination slot, designed to test advanced analytical and logical reasoning skills. Below is a detailed breakdown of the problem, its solution, and key insights to tackle similar questions.
Problem Statement
*“A committee of 5 members is to be formed from 7 men and 6 women. In how many ways can this be done if:
The committee must have at least 3 men.
The chairperson (chosen from the committee) must be a woman, and the vice-chairperson (chosen from the remaining members) must be a man.”*
Solution Steps
Part 1: Committee Composition
The first condition requires the committee to have at least 3 men. This means calculating combinations for three scenarios:
3 men + 2 women
4 men + 1 woman
5 men + 0 women
Calculations:
3M + 2W:
( C(7,3) \times C(6,2) = 35 \times 15 = 525 )
4M + 1W:
( C(7,4) \times C(6,1) = 35 \times 6 = 210 )
5M + 0W:
( C(7,5) \times C(6,0) = 21 \times 1 = 21 )
Total ways for committee composition:
( 525 + 210 + 21 = 756 )
Part 2: Assigning Roles
The second condition introduces role assignments:
Chairperson (must be a woman)
Vice-chairperson (must be a man from the remaining members)
Approach:
Select the committee first (as above).
Assign roles based on the committee’s gender composition.
Case Analysis:
Case 1: 3M + 2W
Chairperson: 2 women → ( 2 ) choices.
Vice-chairperson: 3 men → ( 3 ) choices.
Total ways: ( 525 \times 2 \times 3 = 3150 ).
Case 2: 4M + 1W
Chairperson: 1 woman → ( 1 ) choice.
Vice-chairperson: 4 men → ( 4 ) choices.
Total ways: ( 210 \times 1 \times 4 = 840 ).
Case 3: 5M + 0W
Invalid: No women to be chairperson.
Total ways: ( 0 ).
Final Answer:
( 3150 + 840 = 3990 ) ways.
Key Insights
Break Down Complex Problems: Split multi-step questions into smaller parts (e.g., committee selection vs. role assignment).
Use Combinatorial Logic: Apply ( C(n,k) = \frac{n!}{k!(n-k)!} ) rigorously.
Check Validity of Cases: Some scenarios (e.g., all-male committees) may be invalid due to constraints.
Role Assignments Multiply Possibilities: Each role assignment is a multiplicative factor after committee selection.
Common Mistakes to Avoid
Forgetting to check if a committee composition is valid for role assignments (e.g., no women in a 5M committee).
Overlooking the multiplicative effect of role assignments (e.g., ( 2 \times 3 = 6 ) ways per committee in Case 1).
Miscalculating combinations (e.g., ( C(7,3) = 35 ), not 21).
Conclusion
This problem exemplifies how CAT exams blend combinatorics with conditional logic. Mastery of permutations, careful case analysis, and attention to constraints are critical. Practice similar questions to build speed and accuracy, especially those involving multi-stage role assignments.
Good luck to all CAT aspirants! 🚀
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