The number of permutations of 4 objects chosen from 8 is given by the function
P(n, k) = n!/(n-k)!
where you have n=8 and k=4.
It gives the result P(8, 4) = 8·7·6·5 = 1680.
You can see that this is the result you would get when you consider that the first selection can be any of 8, the next selection any of the remaining 7, the next any of the remaining 6, and the final selection any of the remaining 5. The number of possible arrangements is the product of these numbers of possible selections. (The final number may need to be adjusted if there are unimportant distinctions between the arrangements—as rotations of a circular arrangement, for example.)