Answer: C

Solution:

From the leftmost column, there is no carry into a sixth digit, so $A+B=D\le 9$.

The units column is $B+A=D$, so it also has no carry. The tens column then gives $C+D=D$, hence $C=0$.

Since $A$ and $B$ are distinct nonzero digits, $D=A+B$ can be any digit from $3$ through $9$. For example, $(A,B)=(1,2),$ $(1,3),$ $(2,3),$ $(2,4),$ $(2,5),$ $(2,6),$ $(2,7)$ give $D=3,4,5,6,7,8,9$.

Thus there are $7$ possible values of $D$, and the correct answer is C .