Raymond Barbour
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You have 10 different beer bottles and their 10 labels. You randomly stick labels on bottles. On average how many labels will be on the correct bottle?

This is in a book, Lateral Thinking Puzzles: A Cocktail of Conundrums for Connoisseurs, from Lagoon Books, London.

One. It is far more likely that one will end up on the right bottle than that ten will be on the wrong bottle.

I have always found the explanation rather glib and unhelpful.

In trying to calculate the answer there was one thing I could not calculate: how many ways can x items be arranged so that none are in the correct place?

Does anyone know a formula for this?

Using a spreadsheet I was able to start with x=1 then by solving the puzzle for x bottles I could use the result to work out x+1.

It does turn out that the average is 1. It also turns out that the "explanation" is completely wrong. For even numbers of bottles the number of ways to have one correct is actually 1 way less than the number of ways to have none correct. For odd numbers of bottles it is 1 way more. 1334961 vs 1334960 for 10 bottles.
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did you catch this classy move by the Trump administration?
Six naked eye planets and moon visible on this cold winter evening.
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Meteorite strike here in South Africa.