2014 AMC 12B 第 8 题

The leftmost column gives $A+B=D$ with no carry, so $A+B \le 9$

The second and fourth columns force $C=0$ and no carries anywhere

The leftmost column shows $A+B = D$ with no carry out, so $A+B \le 9.$ Examining the tens and thousands columns (each of the form $C + \text{digit} + \text{carry}$ producing the same digit) forces $C = 0$ and eliminates all carries.

Every column then reduces to $A+B = D,$ with $A, B, C=0$ distinct. Since $A$ and $B$ are distinct positive digits, $D = A+B$ can be any value from $3$ up to $9,$ giving $7$ possibilities, for example $(A,B,C,D) = (1,2,0,3),$ $(1,3,0,4), \ldots,$ $(2,7,0,9).$

Thus, the correct answer is C.