PokéBase - Pokémon Q&A
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Assuming each IV has a 1/32 chance of having each value, there are 32^6 possible values that 6 IVs can have. There are 6 stats that can have the one perfect IV. There are 6 - 1 other stats and 32 - 1 values that they can have (because they are not the perfect IV). So there are 6 * (32 - 1)^(6 - 1) ways to have one perfect IV, and the chance of getting that is 6 * (32 - 1)^(6 - 1) / (32^6), or roughly 1/6.

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