Title: CAT 2021 QA Slot 2: Indian Context Math Problem
Problem Statement (Hypothetical Example):
In a traditional Indian village, there are three communities: Brahmins, Kshatriyas, and Vaishyas. Each community has members who can only marry outside their own community. If the population of Brahmins is 120, Kshatriyas is 150, and Vaishyas is 180, what is the probability that a randomly selected married couple consists of members from different communities? Assume all marriages are between two different communities.
Solution:
Total Possible Marriages:
Since marriages are only between different communities, calculate the total number of valid pairs:
Brahmins × Kshatriyas: (120 \times 150 = 18,000)
Brahmins × Vaishyas: (120 \times 180 = 21,600)
Kshatriyas × Vaishyas: (150 \times 180 = 27,000)
Total Marriages: (18,000 + 21,600 + 27,000 = 66,600)
Total Population:
Total villagers = (120 + 150 + 180 = 450).
Probability Calculation:
The probability of selecting a couple from different communities is the ratio of valid marriages to all possible pairs (including same-community pairs, which are invalid):
Total Possible Pairs (including invalid): (\binom{450}{2} = \frac{450 \times 449}{2} = 101,025).
Valid Probability: (\frac{66,600}{101,025} \approx 0.658) or 65.8%.
Alternative Approach (Simplified):
Probability = 1 - Probability of same-community marriages:
Same-community pairs: (\frac{120 \times 119}{2} + \frac{150 \times 149}{2} + \frac{180 \times 179}{2} = 7,140 + 11,175 + 16,110 = 34,425).
Probability = (1 - \frac{34,425}{101,025} \approx 0.658).
Key Takeaways:
Combinatorial Reasoning: Use combinations to count valid vs. invalid pairs.
Contextual Awareness: Leverage real-world scenarios (Indian marriage laws) to frame abstract math problems.
Efficiency: Simplify calculations by recognizing reciprocal probabilities (e.g., 1 - invalid probability).
Final Answer:
The probability is approximately 65.8%.
Let me know if you need further clarification or a different problem type!
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