Permutations Involving 8 Elements

Probability Level 5

How many permutations τ \tau on 8 elements are there such that ττ \tau \circ \tau is the identity permutation?

Details and assumptions

You may think of a permutation on 8 elements as a way to shuffle a deck of 8 cards. The identity permutation would correspond to the shuffle where we do nothing (hence it stays the same = identity). If τ\tau corresponds to only exchanging the top 2 cards, then ττ \tau \circ \tau would return the deck to it's unchanged state.

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