Answer
There are 720 different arrangements.
Work Step by Step
1st letter: 6 choices.
2nd letter: 5 choices.
3rd letter: 4 choices.
4th letter: 3 choices.
5th letter: 2 choices.
6th letter: 1 choices.
Using the Multiplication Rule of Counting (see page 298) with fatorial:
$6\times5\times4\times3\times2\times1=6!=720$