Derangements of $5$ elements

#derangements#$D_5$#computation

Statement

Compute $D_5$ , the number of derangements of $5$ elements.

Hint

$D_5=5!\sum_{k=0}^5\frac{(-1)^k}{k!}$ .

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Compute. $D_5=120(1-1+1/2-1/6+1/24-1/120)=120(1/2-1/6+1/24-1/120)$ .

$\frac12-\frac16+\frac1{24}-\frac1{120}=\frac{60-20+5-1}{120}=\frac{44}{120}$ .

So $D_5=120\cdot44/120=44$ .

Conclusion. $D_5=44$ .