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Division and Distribution of Objects

(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_rthings (a₁ ≠ a_j)

= ⁿC_a₁. ^n-a¹C_a₂. ^n-a¹^-a²C_a_r. = 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 .

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