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Combinations


Types of Combinations
Simple Combinations
Partition into Groups
Weak Composition
Strong Composition
Distribution into Cells


Combinations: Selecting and Grouping without Order

Combinations focus on selection where order doesn't matter. Unlike permutations where arrangement sequence is crucial, combinations only care about which elements are chosen, not how they're arranged.

This page covers five key combination scenarios: Basic Combinations (selecting unordered subsets), Partition into Groups (dividing elements into unlabeled groups), Distribution into Cells (assigning elements to labeled containers), Weak Composition (distributing identical units with empty cells allowed), and Strong Composition (distributing identical units requiring non-empty cells).

Each type serves different counting needs, from team selection to resource allocation. Understanding when order matters versus when it doesn't is fundamental to choosing between combinations and permutations.

Mastering these combination types equips you to handle selection and distribution problems across various mathematical and practical contexts. The challenge is identifying which scenario matches your specific problem.




Types of Combinations

In combinatorics, combinations represent ways to select or distribute items without considering order, in contrast to permutations where order matters. Building on our earlier comparison between combinations and permutations, we now distinguish between different types of combinations using key questions: whether the objects are distinct or identical, whether we are selecting a subset or distributing all objects, and whether the containers (cells) are labeled or can remain empty.

Combination Scenarios

ScenarioObjects Distinct Or Identical?Select Subset Or Distribute?Cells Labeled Or Unlabeled?Empty Cells Allowed?FormulaExample
Simple CombinationDistinctSelectN/AN/A(nr)\binom{n}{r}Choose 3 fruits out of 10 different types
Partition into GroupsDistinctDistributeUnlabeledTypically non!n1!n2!…nk!\frac{n!}{n_1!n_2!\dots n_k!} (Stirling numbers for equal partitions)Split 12 players into 3 unlabeled teams of 4
Weak CompositionIdenticalDistributeLabeledYes(n+r−1r−1)\binom{n+r-1}{r-1}Distribute 7 identical candies to 3 children (some may get none)
Strong CompositionIdenticalDistributeLabeledNo(n−1r−1)\binom{n-1}{r-1}Distribute 7 identical candies to 3 children (each gets at least 1)
Distribution into CellsDistinctDistributeLabeledYesrnr^nAssign 5 employees to 3 different projects

Conclusion:
Understanding these different combination scenarios is essential because each situation leads to a different counting formula. Recognizing the correct type of combination ensures accurate problem-solving in probability, statistics, and many applied fields.

Simple Combinations

Simple Combination works when selecting items from a collection where order doesn't matter and no repetitions are allowed. Classic examples include choosing team members from a group or selecting lottery numbers.

Property Simple Combination
Order matters ✗
Objects distinct or identical? Distinct
Select subset or distribute? Select subset
Cells labeled or unlabeled? N/A
Empty cells allowed? N/A

Simple Combination — Examples:
Choosing committee members from a club, selecting lottery numbers, picking a set of books to borrow from a library, forming a group of students for a project, selecting ingredients for a recipe from available options.

Common notations:
•(nr)\binom{n}{r} - "n choose r" (most common in modern texts)
•C(n,r)C(n,r) - C for "combinations"
•CnrC_n^r - alternative C notation
•nCr_nC_r - subscript/superscript form

The (nr)\binom{n}{r} notation is the most widely used in modern mathematics, but you'll often see C(n,r)C(n,r) in introductory texts and nCr_nC_r on calculators.


Learn more about combinatorial notations and symbols here →

Formula:
The total number of simple combinations is
                                                 (nr)=n!r!(n−r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}

If nn is the total number of distinct items and rr is the number of items to select, then order doesn't matter and repetition isn't allowed.

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Simple Combinations Examples

n = 3 items choose r = 2 C(3,2) = 3 3!/(2!×1!) = 3 n = 4 items choose r = 2 C(4,2) = 6 4!/(2!×2!) = 6 n = 5 items choose r = 3 C(5,3) = 10 5!/(3!×2!) = 10 Combination: Select r from n, order doesn't matter Formula: C(n,r) = n!/(r!×(n-r)!) - Choose items, {A,B} = {B,A}

Partition into Groups

Partition into Groups applies when dividing distinct items into unlabeled subsets where only grouping matters, not the order within groups or names of the groups. Classic examples include dividing students into teams or splitting items into categories.

Property Partition into Groups
Order matters ✗
Objects distinct or identical? Distinct
Select subset or distribute? Distribute
Cells labeled or unlabeled? Unlabeled
Empty cells allowed? Typically no

Partition into Groups — Examples:
Dividing students into study groups, splitting employees into project teams, grouping tasks into phases of a project, dividing guests into tables at an event, forming clusters of data points in analysis.

Notation:
Partition into Groups doesn't have a standard symbolic notation like (nr)\binom{n}{r} for simple combinations.

Partitions are typically just described in words or with set notation like:

"Partition of set S into k parts"
{A₁, A₂, ..., Aₖ} where A₁ ∪ A₂ ∪ ... ∪ Aₖ = S

Unlike other combinations, permutations, etc., partitions don't have a universally recognized compact notation symbol.


Learn more about combinatorial notations and symbols here →

Formula:
The total number of partitions into groups is
                                                 {A₁, A₂, ..., Aₖ}=n!n1!n2!…nk!\frac{n!}{n_1!n_2!\dots n_k!}

If nn is the total number of distinct items to partition into kk groups of sizes n1,n2,…,nkn_1, n_2, \ldots, n_k respectively, where the groups are unlabeled and only the grouping matters.


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Divide 4 distinct items into Group 1 (size 2) and Group 2 (size 2) A B C D Experiment 1 Group 1 A B Group 2 C D Experiment 2 Group 1 A C Group 2 B D Experiment 3 Group 1 A D Group 2 B C Experiment 4 Group 1 B C Group 2 A D Experiment 5 Group 1 B D Group 2 A C Experiment 6 Group 1 C D Group 2 A B Partition into Groups of Known Sizes Formula: 4!/(2! × 2!) = 24/4 = 6 ways • Choose 2 items for Group 1, remaining 2 automatically go to Group 2 • Groups are distinguishable: Group 1 ≠ Group 2

Weak Composition

Weak Composition occures when distributing identical units into labeled containers where some containers can remain empty and only the count in each container matters. Classic examples include distributing identical coins into numbered boxes or allocating identical resources to different departments.


Property Weak Composition
Order matters ✗
Objects distinct or identical? Identical
Select subset or distribute? Distribute
Cells labeled or unlabeled? Labeled
Empty cells allowed? ✓

Weak Composition — Examples:
Distributing identical coins into numbered boxes, allocating identical resources to different departments, assigning identical tasks across several workers. Since this type of experiment is highly specific — focusing only on counts in labeled containers, even allowing some to remain empty — it is largely self‑explanatory and not as easy to find in common real‑world situations.

Notation:
The notation for Weak Composition is:

((nr))\left(\binom{n}{r}\right) or (((nr)))\left(\left(\binom{n}{r}\right)\right)

This represents the number of ways to place n identical objects into r labeled bins, where empty bins are allowed. It's also written as (n+r−1r−1)\binom{n+r-1}{r-1} but the double parentheses notation specifically indicates weak compositions.

Learn more about combinatorial notations and symbols here →

Formula:
The total number of weak compositions is
                                                 ((nr))=(n+r−1r−1)\left(\binom{n}{r}\right)=\binom{n+r-1}{r-1}

If nn is the total number of identical items to distribute into rr labeled containers, where empty containers are allowed and only the count in each container matters.


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Distribute 3 identical coins into 3 labeled boxes ¢ ¢ ¢ Way 1: (3,0,0) Box 1 Box 2 empty Box 3 empty Way 2: (2,1,0) Box 1 Box 2 Box 3 empty Way 3: (2,0,1) Box 1 Box 2 empty Box 3 Way 4: (1,2,0) Box 1 Box 2 Box 3 empty Way 5: (1,1,1) Box 1 Box 2 Box 3 Way 6: (1,0,2) Box 1 Box 2 empty Box 3 Way 7: (0,3,0) Box 1 empty Box 2 Box 3 empty Way 8: (0,2,1) Box 1 empty Box 2 Box 3 Way 9: (0,1,2) Box 1 empty Box 2 Box 3 Way 10: (0,0,3) Box 1 empty Box 2 empty Box 3 Weak Composition Formula: C(n+k-1, k-1) = C(3+3-1, 3-1) = C(5,2) = 10 ways • Boxes can be empty • Order of boxes matters • Only counts matter, not which specific coin goes where

Strong Composition

Strong Composition works when distributing identical units into labeled containers where every container must receive at least one unit and only the count in each container matters. Classic examples include distributing identical items to departments where each department must get something or allocating identical resources ensuring no group is left empty.

Property Strong Composition
Order matters ✗
Objects distinct or identical? Identical
Select subset or distribute? Distribute
Cells labeled or unlabeled? Labeled
Empty cells allowed? ✗

Strong Composition — Examples:
Distributing identical items to departments where each department must receive at least one, allocating identical resources among teams ensuring none are left empty, dividing identical tasks across several workers with no one unassigned. Like weak composition, this type of experiment is quite specific — it focuses on counts in labeled containers with no empties allowed — making it largely self‑explanatory and less common in everyday situations.



Notation:
The notation for Strong Composition is:

⟨(nr)⟩\left\langle\binom{n}{r}\right\rangle or ⟨⟨(nr)⟩⟩\left\langle\left\langle\binom{n}{r}\right\rangle\right\rangle

This represents the number of ways to place n identical objects into r labeled bins where each bin must contain at least one object. It's also written as (n−1r−1)\binom{n-1}{r-1} but the angle brackets notation specifically indicates strong compositions.


Learn more about combinatorial notations and symbols here →

Formula:
The total number of strong compositions is
                                                 ⟨(nr)⟩=(n−1r−1)\left\langle\binom{n}{r}\right\rangle=\binom{n-1}{r-1}

If nn is the total number of identical items to distribute into rr labeled containers, where each container must receive at least one item and only the count in each container matters.


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Distribute 5 identical coins into 3 labeled boxes (each box must get at least 1) ¢ ¢ ¢ ¢ ¢ Way 1: (3,1,1) Box 1 Box 2 Box 3 Way 2: (2,2,1) Box 1 Box 2 Box 3 Way 3: (1,3,1) Box 1 Box 2 Box 3 Way 4: (1,2,2) Box 1 Box 2 Box 3 Way 5: (1,1,3) Box 1 Box 2 Box 3 Way 6: (2,1,2) Box 1 Box 2 Box 3 Strong Composition Formula: C(n-1, k-1) = C(5-1, 3-1) = C(4,2) = 6 ways • No boxes can be empty • Each box must get at least 1 unit

Distribution into Cells

Distribution into Cells works when assigning each distinct item to a specific labeled container, creating a mapping that shows which container each item goes to. Classic examples include assigning students to different classrooms or placing different files into labeled folders.

Property Distribution into Cells
Order matters ✗
Objects distinct or identical? Distinct
Select subset or distribute? Distribute
Cells labeled or unlabeled? Labeled
Empty cells allowed? ✓

Distribution into Cells — Examples:
Assigning students to classrooms, placing different files into labeled folders, allocating distinct products to specific storage bins, or assigning employees to designated offices. This setup focuses on mapping each unique item to a labeled container, making the outcome a clear one‑to‑one assignment.

Notation:

Distribution into Cells doesn't have a standard symbolic notation like (nr)\binom{n}{r} for simple combinations. It's typically just described as "functions from set A to set B" or "mappings" in mathematical notation, but there's no compact symbol specifically for this concept like there is for combinations or permutations.

Learn more about combinatorial notations and symbols here →

Formula:
The total number of distributions into cells is
                                                 rnr^n

If nn is the total number of distinct items to assign and rr is the number of labeled containers, where each item can go into any container and containers can be empty.


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1 2 3 4 r r cells n items Step 1: Placing first (red) ball The red ball can be placed in any of the r cells. Therefore, there are r ways to place the red ball. Total ways for Step 1: r AND Step 2: Placing second (blue) ball The blue ball can be placed in any of the r cells. Therefore, there are r ways to place the blue ball. Total ways for Step 2: r AND ... AND Step n: Placing nth (purple) ball The purple ball can be placed in any of the r cells. Therefore, there are r ways to place the purple ball. Total ways for Step n: r Since all steps are independent experiments and each step has r ways, the total number of ways to distribute n items into r cells is: rn