Find the number of all possible ways of placing 8 different books on a shelf.

It is also     $$^8P_8$$

That is, there are 8 different objects.  In how many ways can you select 8 and order them.

$${\left({\frac{{\mathtt{8}}{!}}{({\mathtt{8}}{\mathtt{\,-\,}}{\mathtt{8}}){!}}}\right)} = {\mathtt{40\,320}}$$

32 positions

$${\mathtt{8}}{!} = {\mathtt{40\,320}}$$

idea: divide the shelf in 8 imaginary slots.
you can place the first book in one of 8 slots, that's 8 possiblities.

now one slot is occupied, so the shelf has only 7 remaining slots. the second book can now be placed in one of 7 slots, that's 7 possiblities.

and so on..

the last book can only be placed in excactly one remaining slot, that's 1 possibility.
total possibilities = 8*7*6*5*4*3*2*1 = 8! (factorial) = 40320