So I've been working on this combinatorics problem for a few hours now and not coming to any concrete solutions. I was hoping someone could help me figure out how to think about this problem so I can solve it. I know it will use some combination of the product and sum rules but haven't been able to figure it out this far. Here is the problem:
A 100-page document is being printed by four printers. Each page will be printed exactly once. The 4 printers print black and white. Suppose that all the pages are black-and-white, but each group of 25 consecutive pages (1-25, 26-50, 51-75, 76-100) must be assigned to the same printer. Each printer can be assigned 0, 25, 50, 75, or 100 pages to print. How many ways are there for the 100 pages to be assigned to the four printers?
I came to the conclusion that it cannot be $5^4$ because we can only print out 100 pages, so we wouldn't have all options such as 100 pages being printed by each printer simultaneously, nor would we have 0 pages being printed by each printer simultaneously. The only way I came to an answer was by attempting to write out all possible combinations that resulted in 100 pages being printed, which I determined to be 31 different ways to print out the 100 pages.
Any guidance to solve this problem would be greatly appreciated.
Thank you all so much.
Best Answer
Assume the printers are bins and that the groups of pages are colored balls with each color corresponding to each group of numbers. For example, red corresponds to 1-25, blue to 26-50 and so on. So there are 4 bins and 4 balls of different colors and you want to assign balls to bins such that any bin can contain any number of balls. You can assign the first ball to any of the 4 bins and so there are 4 ways of assigning the first ball. Having assigned the first ball, you can then assign the second ball in, again, 4 ways. Similarly, the third and the fourth balls can each be assigned in 4 ways. So total numbers of ways is 4 x 4 x 4 x 4 = 256.