In all, there are 8*7*6*5 = 1680 different arrangements.


It is the product of 4 integer factors, in descending order, starting from the biggest factor 8.


The standard mantra to solve this (and many other similar problems) is THIS :


    There are 8 ways to select the first book;

    then there are 7 ways to select the second book from 7 remaining books;

    then there are 6 ways to select the third book from 6 remaining books;

    and finally, there are 5 ways to select the fourth book from 5 remaining books.