Number of Derangements on Finite Set/Examples/7

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Example of Number of Derangements on Finite Set

A correspondent writes $7$ letters and addresses $7$ 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:

\(\ds D_7\) \(=\) \(\ds 7! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \dfrac 1 {4!} - \dfrac 1 {5!} + \dfrac 1 {6!} - \dfrac 1 {7!} }\)
\(\ds \) \(=\) \(\ds 5040 \paren {1 - \dfrac 1 1 + \dfrac 1 2 - \dfrac 1 6 + \dfrac 1 {24} - \dfrac 1 {120} + \dfrac 1 {620} - \dfrac 1 {5040} }\)
\(\ds \) \(=\) \(\ds 5040 - 5040 + 2520 - 840 + 210 - 42 + 7 - 1\)
\(\ds \) \(=\) \(\ds 1854\)

$\blacksquare$


Sources