If $\text{A}$ and $\text{B}$ are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is
\[\begin{tabular}[t]{cccc} 9 & 8 & 7 & 6 \\ & A & 3 & 2 \\ & B & 1 \\ \hline  \end{tabular}\]

$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{depends on the values of A and B}$
The minimum possible value of this sum is when $A=B=1$, which is \[9876+132+11=10019\]
The largest possible value of the sum is when $A=B=9$, making the sum \[9876+932+91=10899\]
Since all the possible sums are between $10019$ and $10899$, they must have $5$ digits.
$\boxed{\text{B}}$