For the first position, I have four choices, since any of the books can go there. For the second position, I have three choices, since I can't repeat the book I already put in the first position. For the third position, I have two choices, since I can't repeat the books I already put in the first and second positions. For the fourth position, I have one choice, since there is only one book left. So the total number of ways to arrange four books is 4 x 3 x 2 x 1, which is also called 4 factorial and written as 4!. I can use a calculator or do the multiplication by hand to get 4! = 4 x 3 x 2 x 1 = 24. 
In how many ways can I arrange four different books on a shelf?