math
posted by lizzie
Five books are put in three boxes, how many ways are there if:
a) The books are different and the boxes are different?
b) The books are the same and the boxes are different?
c) The books are different and the boxes are the same?
d) The books are the same and the boxes are the same?

PsyDAG
I'm confused as to what you mean by the "same" and "different." Can you rephrase the problem in a new post?

Reiny
Interesting question.
let's call our boxes A,B, and C
and our books 1,2,3,4, and 5
let's do b) first
our books are all the same, so we will just separate them into 3 piles adding to 5
1 1 3
1 2 2
1 3 1
2 1 2
2 2 1
3 1 1
so if our first column is box A, second column is B etc
then there are 6 ways.
d) if the boxes are the same as well, then isn't 1 2 2 the same as 2 1 2?
so there are only 2 distinct ways , namely 1 1 3 and 1 2 2, since the order does not matter.
a) notice in b) we found there are 6 ways to put identical books in 3 distinct boxes.
Now suppose that all the books are different.
Couldn't these 5 books now be arranged in 5! or 24 ways, without changing the count in the 3 distinct boxes?
so with all books different and boxes different, the number of ways would be 6x24 or 144 ways.
That leaves c)
see if you can reason it out.
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