12. If: (1) $a, b, c, d$ all belong to $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ is the smallest value among $a, b, c, d$,

then, the number of different four-digit numbers $\overline{a b c d}$ that can be formed is
28
12. 【Analysis and Solution】When $\overline{a b c d}$ has exactly 2 different digits, a total of $\mathrm{C}_{4}^{2}=6$ different numbers can be formed. When $\overline{a b c d}$ has exactly 3 different digits, a total of $\mathrm{C}_{3}^{1} \mathrm{C}_{2}^{1} \mathrm{C}_{2}^{1}+\mathrm{C}_{2}^{1} \mathrm{C}_{2}^{1}=12+4=16$ different numbers can be formed. When $a b c d$ has exactly 4 different digits, a total of $\mathrm{P}_{3}^{3}=6$ different numbers can be formed.
$\therefore$ In total, $6+16+6=28$ (numbers) can be formed.