Derangements of $3$ elements

#derangements#$D_3$#list

Statement

List the derangements of $\{1,2,3\}$ and count them.

Hint

A derangement of $3$ is a $3$ -cycle.

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Derangements. $(123)$ and $(132)$ . Both are $3$ -cycles.

Count. $2$ .

Check. $D_3=6(1-1+1/2-1/6)=6\cdot2/6=2$ . ✓

Conclusion. $D_3=2$ .