Nine Bottles

There are nine bottles, each containing a different type of liquid, but the labels have all fallen off. Knowing nothing about the contents, a passerby reapplies the labels at random. What is the expected number of correctly labeled bottles? (That is, if the labels are randomly assigned many times, what is the average number of correct labels?)

It might be easier to try the problem for smaller numbers of bottles first.

Source: Internet newsgroup rec.puzzles, Used as Ken's POTD 5/16/94.

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