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Divisibility by 10

In how many w...

Question

In how many ways we can put $5$ writings into $5$ corresponding envelopes so that no writing go to the corresponding envelope $?$

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Solution

The problem is the number of dearragements of $5$ digits.
This is equal to $5! (\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!}) = 44.$

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