From 1882 intuition to modern proofs: we explore how just two random permutations almost surely generate the full symmetric group S_n (or the alternating group A_n) as n grows. We trace Dixon’s 1969 breakthrough showing the probability tends to 1, the famously slow convergence explained by a 1/n error term tied to shared fixed points, and Babai’s 1989 refinement using the classification of finite simple groups. Along the way we’ll connect parity, fixed points, and the deep structure that turns randomness into almost-sure universal reach in permutation groups.

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