Find the number of permutations of the first 8 letters of the alphabet taking 5 letters at a time? !

The first letter can be any one of 8.  For each of those . . .
The second letter can be any one of the remaining 7.  For each of those . . .
The third letter can be any one of the remaining 6.  For each of those . . .
The fourth letter can be any one of the remaining 5.  For each of those . . .
The fifth letter can be any one of the remaining 4.

The total number of possibilities is  (8 x 7 x 6 x 5 x 4) = 6,720 .
(That's  8! / 3! .)

Note:
If you're allowed to use the same letter more than once,
then there are 8 choices for each of the 5 letters.
The total number of possibilities then is              (8 x 8 x 8 x 8 x 8) = 32,768 .
(That's 8⁵  or  2¹⁵ .)