Answer
80,318,101,760 user names are possible for the local area network.
Work Step by Step
For each letter there are 26 choices (repetition of letters is allowed).
Using the Multiplication Rule of Counting (see page 298):
$26\times26\times26\times26\times26\times26\times26=26^{7}$.
For the last character (a digit) there are 10 choices.
Using the Multiplication Rule of Counting (see page 298):
$26^{7}\times10=80,318,101,760$
