Division and Distribution of Objects
Division and distribution of objects are fundamental concepts in combinatorics that deal with the allocation of items or elements from a set among various categories or recipients. These concepts are often encountered in realworld scenarios involving resource allocation, task assignment, and more.
Division of Objects:

Definition: Division of objects refers to the process of dividing a set of objects into distinct groups or categories based on specific conditions or constraints. It can involve allocating items into partitions, classes, or containers.

Examples:
 Distributing 12 identical candies among 3 children so that each child receives an equal number.
 Dividing a group of people into teams of varying sizes while ensuring certain individuals are on the same team.

Approach:
 Division problems often involve partitioning the total number of objects into distinct groups.
 The key is to find the number of ways to distribute objects while satisfying the given conditions, such as equal division or specific groupings.
Distribution of Objects:

Definition: Distribution of objects involves the allocation of a set of items to a group of recipients, such as individuals, entities, or containers. The goal is to determine how many ways the items can be assigned to the recipients, considering various constraints.

Examples:
 Distributing 5 different prizes among a group of 10 students where each student can receive at most one prize.
 Allocating 20 identical pens to 4 classrooms in such a way that each classroom receives a different number of pens.

Approach:
 Distribution problems often require the use of combinations, permutations, or other combinatorial techniques, depending on the constraints.
 You need to consider factors like identical items, the number of recipients, and any restrictions on the distribution.
Key Concepts:
 Unequal Division: In unequal division problems, objects are divided into groups of unequal size. The number of ways to divide n distinct objects into r unequal groups containing ${a}_{1}$, ${a}_{2}$, ${a}_{3}$, ......, ar things (${a}_{1}$ ≠ aj) is
nC${a}_{1}$. n${a}_{1}$C${a}_{2}$. n${a}_{1}$${a}_{2}$C${a}_{r}$. = n!/${a}_{1}$!${a}_{2}$!${a}_{3}$!...${a}_{r}$!
Here ${a}_{1}$ + ${a}_{2}$ + ${a}_{3}$ + ...... + ${a}_{r}$ = n.
 Equal Division: In division problems, achieving equal division or distribution is a common objective, and you should use appropriate combinatorial methods to determine the number of ways this can be done.
In equal division problems, objects are divided into groups of equal size. The number of ways to divide m × n distinct objects equally into n groups is (mn)!/(m!)n n!.
The number of ways to distribute m × n different objects equally among n persons (or numbered groups) is (mn)!/(m!)n n!.
 Distribution with Conditions: In distribution problems with conditions, objects are distributed among individuals with specific conditions.
For example, in distributing n different objects among r distinct groups such that all of them must get at least one, the number of ways to arrange 7 different things to 3 people, such that all of them must get at least one is
nCr × ${(r1)}^{(nr)}$.
 Permutations and Combinations: Depending on the problem's nature, you may need to use permutations for ordered distribution or combinations for unordered distribution.
 Resource Optimization: These concepts are applicable in resource allocation, task assignment, and decisionmaking processes, where efficient distribution is critical.
Applications:
 Division and distribution problems are frequently encountered in areas like project management, logistics, allocation of resources, fair distribution of assets, and planning.
 They are used in scenarios involving the allocation of goods, tasks, responsibilities, or awards among different entities or individuals.
Examples
 Example 1: In how many ways can 10 distinct objects be divided into 3 groups of unequal size?
The number of ways to divide 10 distinct objects into 3 groups of unequal size is
10C3. 7C4. 3C1 = 1200 .
 Example 2: In how many ways can 12 distinct objects be divided equally into 4 groups?
The number of ways to divide 12 distinct objects equally into 4 groups is
(12 × 11 × 10 × 9)/(4 × 3 × 2 × 1) = 495 .
 Example 3: In how many ways can 7 different things be distributed among 3 people such that all of them must get at least one?
The number of ways to arrange 7 different things to 3 people, such that all of them must get at least one is 3! × (${2}^{4}$  2) = 42.