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
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): The misaddressed letters: $130$