Answer to Question #100370 in Combinatorics | Number Theory for Aaron

Question

The six graders at Washington middle school research the history of their city this day still gave a presentation to the other students at the school if they were 64 six skaters list all the ways they could have been divided equally into groups of 10 or fewer students

Answer 1

The number of ways to divide n different objects into x groups of size a each, y groups of size b each, and z groups of size c each is equal to  n!/( (a!)^x(b!)^y(c!)^z.x!.y!.z! )

but here we have to divide each group in equal sizes so our formula gets reduced to

n!/( (a!)^x.x! )

In the question we are given 64 students and we have to divide them into equal groups of strength less than or equal 10

So, we can divide the 64 student into 64 groups of 1 student or 32 groups comprising of 2 students in each group or 16 groups comprising of 4 students in each group or 8 groups comprising of 8 students in each group.

And after adding all the number of ways we will get the answer

64!/((1!)⁶⁴.64!) + 64!/((2!)³².32!) + 64!/((4!)¹⁶.16!) + 64!/((8!)⁸.8!)