In the given operation, the letters $a, b$, and $c$ represent distinct digits and are different from 1. Determine the values of $a, b$, and $c$.

$$
\begin{array}{r}
a b b \\
\times \quad c \\
\hline b c b 1
\end{array}
$$
Since the product of $b$ by $c$ ends in 1, then $b \times c$ can be 21 or 81, and therefore $3 \times 7$ or $9 \times 9$. The only possibility to write the product of two distinct numbers less than 10 is $21=3 \times 7$. Thus, we have only two possible cases.

1st Case: If $b=7$ and $c=3$, we should have

$$
\begin{array}{r}
a 77 \\
\times \quad 3 \\
\hline 7371
\end{array}
$$

but this is impossible, since $\frac{7371}{3}=2457$ has four digits.

2nd Case: If $b=3$ and $c=7$, we have
and, since $\frac{3731}{7}=533$, necessarily $a=5$.

Therefore, the only possibility is $a=5, b=3$ and $c=7$.