(With fixed number of objects in each group)

(i)     Into groups of unequal size (different number of objects in each group)

(a)    Number of ways, in which n distinct objects can be divided into r unequal groups containing a₁, a₂, a₃, ......, a_r

       things (a₁ ≠ a_j)

               = ⁿC_a1. ^n-a1C_a2. ^n-a1-a2C_ar. = n!/a₁!a₂!a₃!...a_r!

               Here a₁ + a₂ + a₃ + ...... + a_r = n.

(b)    Number of ways in which n distinct objects can be distributed among r persons such that some person get a₁ objects,  

        another person get a₂ objects ......... and similarly someone gets a_r objects = n!r! /a₁!a₂!a₃!...a_r!.

Explanation: Let us divide the task into two parts. In the first part, we divide the objects into groups. In the second part, these r groups can be assigned to r persons in r! ways.

(ii)    Into groups of equal size (each group containing same number of objects)

(a)    Number of ways in which m × n distinct objects can be divided equally into n groups (unmarked) = (mn)!/(m!)ⁿ n!.

(b)    Number of ways in which m × n different object can be distributed equally among n persons (or numbered groups)

       = (number of ways of dividing) × (number of groups)! = (mn)!n!/(m!)ⁿ n! = (mn)!/(m!)ⁿ .