10. Color the 6 regions $A, B, C, D, E, F$ in Figure 1, with each region being colored with 1 color, and no two adjacent regions having the same color. If there are 4 colors available, then there are $\qquad$ different coloring schemes.
10. 96
$A, B, C$ have different colors from each other. They have $4 \times$ $3 \times 2=24$ ways of coloring.
For any one of these (let's assume $A-a \quad B-b \quad C-c$. The other color is $d$), then $D$ can be colored with $b, d$, $E$ can be colored with $c, d$, and $F$ can be colored with $a, d$.
Since $D, E, F$ are adjacent to each other, at most 1 can be colored with $d$, so there are $1+3=4$ ways of coloring, thus the total number of different coloring methods is $4 \times 24=96$.