  In the classical paper of Dvoretzky-Erd\H{o}s, asymptotics for the expected
value and the variance of the number of distinct sites visited by a Simple
Symmetric Random Walk were calculated. Here, these results are generalized for
Random Walks with Internal States. Moreover, both weak and strong laws of large
numbers are proved. As a tool for these results, the error term of the local
limit theorem in of Kr\'amli and Sz\'asz is also estimated.
