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Close! You swapped an operation there: it'd be (2^8 button possibilities) ^ (5 minutes * 60 FPS), or about 10^43348.

A bit of a bigger space to search ;)

Only 2^7 button possibilities on a real controller, because you can't press up and down, or left and right, at the same time.

Worn controllers can allow it to happen, however, so I don't think it's worth discounting.