put 24 different books on 4 shelves, each shelf has at least one book(The books are put next to each other). How many ways?
The right answer is C(23,20)*24!, meaning the first divide the 24 slots into 4 groups and each group has 1 slot, we get C(23,20), then put 24 books into these 24 slots in order we get 24!. Therefore the result is C(23,20)*24!.
My answer is C(24,4)*4^20. I first choose 4 books and put them onto each shelf, I get C(24,4), then just put the rest 20 books which ever shelf, it should be 4^20. So my answer is C(24,4)*4^20.
Can anyone tell why my answer is wrong?
Best Answer
You are overcounting. Cases where the first shelf get books $a,b$ are counted once with $a$ among the first four and again with $b$ among the first four. You are also undercounting by not accounting for the order of books on each shelf