Derangements of $4$ elements

Derangements.
- $(12)(34)$
- $(13)(24)$
- $(14)(23)$
- $(1234)$ — check: $1\to2$, $2\to3$, $3\to4$, $4\to1$; no fixed points. ✓
- $(1243)$ — check: $1\to2$, $2\to4$, $4\to3$, $3\to1$. ✓
- $(1324)$ — check: $1\to3$, $3\to2$, $2\to4$, $4\to1$. ✓
- $(1342)$ — check: $1\to3$, $3\to4$, $4\to2$, $2\to1$. ✓
- $(1423)$ — check: $1\to4$, $4\to2$, $2\to3$, $3\to1$. ✓
- $(1432)$ — check: $1\to4$, $4\to3$, $3\to2$, $2\to1$. ✓

Count. $3$ double transpositions + $6$ $4$-cycles = $9$.

Check with formula. $D_4=4!(1-1+1/2-1/6+1/24)=24\cdot9/24=9$. ✓

Conclusion. $9$ derangements.