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📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 Now fix a pair of layers. Each contains 3 genera chosen from 6. We want the number of such pairs that share exactly 2 genera.

📌 Instead of fixing layers, compute probability over all possible pairings. But since the composition of genera per layer is not given, assume each layer independently selects 3 genera uniformly at random from 6 (modeling diversity). Then compute expected probability, but per math olympiad style, assume deterministic but general: better to count total assignments and favorable pairs.

📌 But again, for fixed structure, suppose we compute expected number of such pairs over all configurations? Too complex.

📌 Better: the problem likely assumes that each layers selection is arbitrary but fixed, and we want a general probability under uniform random choice of layers and uniform growth. But standard interpretation: suppose each layers genus set is chosen independently and uniformly at random from all 3-element subsets of 6 genera. Then compute the probability that two independently chosen such layers share exactly 2 genera.

📌 Let $L_1, L_2$ be two randomly chosen 3-element subsets of a 6-element set.

📌 The third element of $L_2$ must be from outside $L_1$, so 3 choices.

📌 Since this holds for any $L_1$ and by symmetry, the desired probability is:

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