Expert: Ahmed Salami - 1/26/2006

Question
this is a tough one but here goes. 10 books need to be arranged in 4 shelves. any shelf can be empty (as in all books being on one shelf or split between 2 shelves). how many different ways are there to sort the books? you must take into acount the order they are in on the shelves also. the only thing i've found so far is 10! times 4 which gives every possibilty if all books were all on one shelf, times 4 for the 4 shelves. but i know there is alot more. thank you in advance and good luck. im really stumped.

Answer
Hi Charles,
Sorry for the time it took. Technical problem with the site.
A good starting note is the fact that if an event can
happen in m ways and another one can happen in n ways,
then the number of ways they can happen together would
be (mxn) ways.
We'll start with the consideration that the books are
all the same i.e the order of arrangement doesn't come
in.
The first book can be placed in any of the four shelves
giving us four choices for it. Each of the remaining
nine books has four choices as well, giving us a total
of 4x4x4x4x4x4x4x4x4x4 = 4^10 ways
Now considering the order in which the books are
arranged, the number of different arrangements of 10
different books is 10! i.e 10x9x8x....x2x1
And so the number of different arrangements possible
with 10 books in 4 different shelves would be
(4^10).10!

I hope i have helped. You can always get back to me.
Regards.