Number of Derangements on Finite Set/Examples/10

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Example of Number of Derangements on Finite Set

A correspondent writes $10$ letters and addresses $10$ envelopes, one for each letter.

In how many ways can all the letters be placed in the wrong envelopes?


Solution

From Number of Derangements on Finite Set:

\(\ds D_{10}\) \(=\) \(\ds 10! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \dfrac 1 {4!} - \dfrac 1 {5!} + \dfrac 1 {6!} - \dfrac 1 {7!} + \dfrac 1 {8!} - \dfrac 1 {9!} + \dfrac 1 {10!} }\)
\(\ds \) \(=\) \(\ds 3 \, 628 \, 800 \paren {1 - \dfrac 1 1 + \dfrac 1 2 - \dfrac 1 6 + \dfrac 1 {24} - \dfrac 1 {120} + \dfrac 1 {620} - \dfrac 1 {5040} + \dfrac 1 {40320} - \dfrac 1 {362880} + \dfrac 1 {3 \, 628 \, 800} }\)
\(\ds \) \(=\) \(\ds 3 \, 628 \, 800 - 3 \, 628 \, 800 + 1 \, 814 \, 400 - 604 \,800 + 151 \, 200 - 30 \, 240 + 5040 - 720 + 90 - 10 + 1\)
\(\ds \) \(=\) \(\ds 1 \, 334 \, 961\)

$\blacksquare$


Sources