Subfactorial/Examples/5

Example of Subfactorial

$!5 = 44$


Proof

\(\ds !5\) \(=\) \(\ds 5! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \dfrac 1 {4!} - \dfrac 1 {5!} }\) Definition of Subfactorial
\(\ds \) \(=\) \(\ds 120 \paren {1 - \dfrac 1 1 + \dfrac 1 2 - \dfrac 1 6 + \dfrac 1 {24} - \dfrac 1 {120} }\) Examples of Factorial
\(\ds \) \(=\) \(\ds 120 \paren {\dfrac {60 - 20 + 5 - 1} {120} }\)
\(\ds \) \(=\) \(\ds 44\)

$\blacksquare$


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $44$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $44$
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