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