I have 5 cards. Card labeled i starts in position i. So, card 1 is first, card 2 is second, card 3 is third, etc. I shuffle them. This gives me a random permutation of the cards. What is the probability that none of my cards are now in the same position they started. In other words: what is the probability that (card 1 is not in spot 1 AND card 2 is not in spot 2 AND card 3 is not in spot 3 AND card 4 is not in spot 4 AND card 5 is not in spot 5)?
2026-04-03 06:22:16.1775197336
shuffling card and permutation
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By symmetry, card 1 is just as likely to be in positions 2,3,4, or 5. So we only need to count the permutations like ${*}1{*}{*}{*}$ and multiply by four.
Assume card 1 is in position 2. There are two ways that, in addition, card 2 ends up in position 1: $21453$ and $21534$. There are 3 ways card 3 ends up in position 1: $31254$, $31524$, and $31452$, and similarly for card 4 or 5 being in position 1. To recap, there are $2+3+3+3=11$ ways with card 1 in position 2.
Multiplying by four shows that the total is 44, so the probability is $44/120=.36666$.