PokéBase - Pokémon Q&A
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In Gen 6, the chances of finding a shiny were lowered to 1 in 4,096.
In a horde battle, there's a 5 in 4096 chance of getting a shiny, since you're encountering 5 Pokemon instead of one.
And if you have the Shiny Charm the chances would be 5 in 1365.3.

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This is wrong, but admittedly by a very small margin.
The chance of finding a shiny in a horde is not equal to 5/4096, rather 1 - (4095/4096)^5. While this doesn't have a huge impact, it still is a calculation error. Same concept applies to the Shiny Charm.
I also admit, that the majority of people who read this will think that I'm off my rockers, but you need to cover basic probability first. ;>