I am very new to combinatorics and want to know how many ways some number of pupils (p) can fit in some number of classes (c) each of the same size (s).
I tried to solve it below using this post:
Say we have 300 pupils that need to be separated into 10 classes of 30.
There are 300! ways to line the students up.
Line all 300 students up and for each 30 students place them in a new class. There are two ways to move students around and achieve the same set of classes:
(1) Move students around within each section of 30. One class can be rearranged in 30! ways. As there are 10 classes, you must raise 30! to 10. This means there is 30!^10 ways to rearrange the students within each class of 30.
(2) Swap entire groups of 30 around. There are 10 classes, so there is 10! ways to do this.
Using both of those methods it means there are 30!^10 * 10! combinations of lined up students that result in the same outcome of classes
This means I divide 300! by 30!^10 * 10!, which means the formula is p! / (s!^c * c!). Obviously this is wrong.
Could someone please explain what I did wrong, what the correct way to do this is, and what the answer is please? All help is appreciated, thanks!