Overview

Many counting problems ask us to split objects into groups.

For example:

• Divide $10$ students into two teams of $5$
• Split $12$ people into groups of sizes $3,4,5$
• Separate objects into groups where the groups may or may not be labelled
• Divide objects with repeated types into smaller collections

The main challenge is deciding whether the groups are labelled.

If the groups have names, such as Team A and Team B, then the groups are labelled.

If the groups have no names, then the groups are unlabelled, and we have to avoid counting the same grouping multiple times.

Motivation

Suppose $8$ students are split into two teams of $4$.

If the teams are called Red Team and Blue Team, then choosing the Red Team automatically determines the Blue Team.

The number of ways is

However, if the two teams are not named, then choosing one group of $4$ first and the other group second counts every split twice.

So the number of unlabelled splits is

This difference between labelled and unlabelled groups is one of the most common sources of mistakes in combinatorics.

Labelled Groups

Suppose we have $n$ distinct objects and want to split them into labelled groups of sizes

where

The number of ways is

This is because we choose which objects go into the first group, then the second group, and so on.

Example 1

How many ways are there to split $10$ students into three labelled groups of sizes $2,3,5$?

The groups are labelled, so the answer is

Equivalently, we can choose:

students for the first group, then

students for the second group, and the remaining $5$ students form the third group.

As such, the answer is

Unlabelled Groups of Different Sizes

If the group sizes are all different, then the sizes themselves identify the groups.

For example, if we split $9$ people into groups of sizes $2,3,4$, then the group of size $2$ cannot be confused with the group of size $3$.

So the number of ways is still

No extra division is needed.

Example 2

How many ways are there to split $9$ students into unlabelled groups of sizes $2,3,4$?

Since the group sizes are different, the answer is

Unlabelled Groups of Equal Sizes

If some groups have the same size and are unlabelled, then we must divide by the number of ways to reorder those equal sized groups.

Example 3

How many ways are there to split $8$ students into two unlabelled groups of $4$?

First choose $4$ students for one group:

But this counts each split twice, because choosing the first group or choosing its complement gives the same final split.

So the answer is

Example 4

How many ways are there to split $12$ students into three unlabelled groups of $4$?

If the groups were labelled, the answer would be

But the three groups are unlabelled, so we divide by

So the answer is

Mixed Group Sizes

Sometimes only some groups have equal sizes.

Example 5

How many ways are there to split $11$ students into unlabelled groups of sizes $2,2,3,4$?

If the groups were labelled, the answer would be

The two groups of size $2$ are indistinguishable, so we divide by

The groups of size $3$ and $4$ are unique by size, so no further division is needed.

So the answer is

Identical Objects and Groups

If the objects are identical and the groups are unlabelled, the problem becomes a partition problem.

For example, splitting $7$ identical balls into $3$ unlabelled nonempty groups means finding unordered positive integer triples whose sum is $7$.

The possible group sizes are

So there are $4$ ways.

Unlike labelled distribution, there is no simple single formula for every unlabelled identical object grouping problem. These are often handled using casework.

Mixtures of Object Types

Some problems involve repeated types of objects.

For example, suppose we have:

• $3$ red balls
• $2$ blue balls
• $1$ green ball

If we want to split them into groups, we must track how many objects of each type go into each group.

These problems are usually solved by:

1. Deciding the possible type counts in each group
2. Counting valid cases
3. Dividing by symmetries if the groups are unlabelled

The key is to avoid treating identical objects as distinct.

Worked Examples

Example 6

How many ways are there to split $6$ students into three unlabelled pairs?

If the pairs were labelled, the answer would be

But the three pairs are unlabelled, so divide by

The answer is

Example 7

How many ways are there to split $10$ students into two unlabelled groups of $3$ and one group of $4$?

If the groups were labelled, the answer would be

The two groups of size $3$ are indistinguishable, so divide by

So the answer is

Example 8

How many ways are there to split $5$ identical balls into two unlabelled non empty groups?

We need unordered positive pairs adding to $5$.

The possibilities are

and

So there are $2$ ways.

Key Contest Insight

When splitting into groups, first decide whether the groups are labelled.

If the groups are labelled, choose objects for each group directly.

If the groups are unlabelled, count as though they were labelled and then divide by the number of ways to permute identical groups.

The most common mistake is forgetting to divide when groups of the same size are unlabelled.

Strategy Tips

• Decide whether the objects are distinct or identical
• Decide whether the groups are labelled or unlabelled
• For distinct objects and labelled groups, use factorial division
• For unlabelled equal sized groups, divide by the factorial of the number of identical groups
• If group sizes are all different, no extra division is needed
• For identical objects into unlabelled groups, use casework or partitions
• Always check whether two counted outcomes are actually the same grouping

Common Pitfalls

• Treating unlabelled groups as labelled
• Dividing by too much when group sizes are different
• Forgetting to divide by $3!$ when forming three identical groups
• Treating identical objects as distinct
• Confusing grouping with distribution to named people
• Missing cases when identical objects are split into unlabelled groups

Practice Problems