Count unordered selections using combinations

Alberta Grade 12 Mathematics

Probability

Count groups when the order of the selected items does not matter

Suppose a school chooses three students for a group. The group containing Ava, Ben, and Kai is unchanged if the names are listed Kai, Ava, and Ben. The chosen students matter, but the listing order does not. This lesson shows how to count such groups. First decide what counts as a different result. Then use combinations to count the possible groups.

What you will learn

Recognize when a selection is unordered.
Use combination notation to count groups of a fixed size.
Evaluate combinations using factorials or a calculator.
Explain why each group is counted once.

1. Review: what makes a different result?

A selection is a choice of items from a larger collection. Before counting, ask what makes one result different from another. If changing the order changes the result, order matters. If the same chosen items still make the same result, order does not matter.

For example, choosing two people for a committee with no assigned roles is unordered. Choosing a captain and a deputy is not: switching the people changes who has each role. A combination counts an unordered selection of a specified number of items.

This lesson is about groups with no different positions or roles. You can test a situation by imagining the chosen items in a different order. If that produces the same group, a combination is suitable.

A combination counts groups, not different listings of the same group.
Check whether the selected items have different roles before choosing a counting method.

2. Combination notation and the rule

Suppose there are

n

available items and you select

r

of them. The number of combinations is written as

\binom{n}{r}

and read “

n

choose

r

.” The value

n

is the total available;

r

is the number selected.

A factorial is a product formed by multiplying a positive whole number by every positive whole number below it. For example,

4!=4\times3\times2\times1

. Also,

0!=1

. Factorials are used in the combination rule.

The rule is

\binom{n}{r}=\frac{n!}{r!(n-r)!}

. The denominator accounts for the different ways one group could be listed. Those listings do not create new groups, so the rule counts each chosen group once. For example, a pair has two possible listings, but it is still one pair.

Use whole-number values with

0\leq r\leq n

. In most group-selection questions,

r

is positive. A calculator may have an

nCr

key; enter the total first and the number selected second. Always interpret the result as a number of groups or selections.

\binom{n}{r}=\frac{n!}{r!(n-r)!}

In $\binom{n}{r}$ , $n$ is the total and $r$ is the number chosen.
Use combinations when listing the same selected items in a different order does not change the outcome.
The result is a count, so it has no physical unit.

3. A small example and a useful relationship

Consider four volunteers: Ana, Bo, Cal, and Dee. To form a pair, Ana and Bo make the same pair as Bo and Ana. Listing all the different pairs gives Ana–Bo, Ana–Cal, Ana–Dee, Bo–Cal, Bo–Dee, and Cal–Dee. There are six pairs.

The formula gives the same count:

\binom{4}{2}=6

. This concrete list helps show why order is ignored. For larger collections, use the rule or a calculator rather than trying to list every group.

There is also a useful way to check a count. Choosing

r

items to include identifies the same group as choosing the other

n-r

items to leave out. Thus

\binom{n}{r}=\binom{n}{n-r}

. For instance, choosing three of five people is equivalent to choosing the two people who are left out.

\binom{n}{r}=\binom{n}{n-r}

Small lists can confirm what a combination counts.
Choosing who is included or who is left out describes the same group.

4. Set up and interpret a combination

Start by describing the group being formed. Identify how many items are available and how many will be chosen. Confirm that there are no assigned roles or positions. Then write the combination and evaluate it.

For example, if a display features four books chosen from ten, the question is about which books are included, not their order. The count is written

\binom{10}{4}

. Using the rule gives

210

possible selections.

A common error is to use a combination whenever a question says “choose.” The word alone is not enough. If the chosen people receive different jobs, switching the people changes the outcome. Use a combination only when the selected items form one group without distinct roles.

To keep your answer clear, state what the number counts. “There are 210 selections of four books” is more informative than giving the number alone.

Translate the situation into a total and a number selected.
Check that order and assigned roles do not create new outcomes.
State the group or selection counted in your final answer.

Worked example

Choose a volunteer team

A community garden has 8 available volunteers. How many different teams of 3 volunteers can be formed?

Selecting a volunteer team

A group of 3 is selected from 8 available volunteers.

Identify the selection
The team has no assigned roles. Changing the order of the names does not make a different team, so this is an unordered selection. There are 8 volunteers available and 3 are chosen.
$n=8,\quad r=3$
Write the combination
Use a combination because it counts groups without treating a new listing order as a new team.
$\binom{8}{3}=\frac{8!}{3!5!}$
Calculate and interpret
Cancel the common factorial factors and multiply. The result counts distinct teams, not ordered lists of volunteers.
$\binom{8}{3}=\frac{8\times7\times6}{3\times2\times1}=56$

Answer: There are 56 different teams of 3 volunteers.

Check: Any one team can be listed in several orders, but it is still the same team. The combination count includes it once.

Worked example

Select books for a display

A library display will feature 4 different books chosen from 10 eligible books. How many different selections are possible?

Selecting books

A group of 4 books is selected from 10 eligible books.

Decide whether order matters
The question asks which four books are included. It does not assign positions to the books, so a different listing order gives the same selection.
Set up the count
There are 10 eligible books and 4 are selected. Use a combination to count the unordered selections.
$\binom{10}{4}$
Calculate
Apply the combination rule and cancel the common factorial factors before multiplying.
$\binom{10}{4}=\frac{10!}{4!6!}=\frac{10\times9\times8\times7}{4\times3\times2\times1}=210$

Answer: There are 210 different selections of 4 books.

Check: Selecting four books to feature is equivalent to identifying the six books not featured. The two descriptions identify the same selection.

Worked example

Form a planning group by choosing who is left out

A youth council has 12 members. A planning group will include 9 members. How many different planning groups can be formed?

Forming a planning group

Choose the 3 members left out from 12; the other 9 form the group.

Confirm the group is unordered
Every member has the same role in the planning group. Changing the order in which the names are written does not create a different group.
Count the members left out
Instead of selecting 9 members to include, identify the 3 members left out. Each choice of 3 members left out determines exactly one planning group, so the count is unchanged.
$\binom{12}{9}=\binom{12}{3}$
Evaluate
Calculate the smaller combination. This gives the number of possible planning groups.
$\binom{12}{3}=\frac{12\times11\times10}{3\times2\times1}=220$

Answer: There are 220 different planning groups.

Check: Each group of 9 has exactly 3 members outside it. Choosing those 3 members identifies one and only one group of 9.

Common mistakes and how to avoid them

Counting each listing order as a different group.

Correction: For an unordered selection, count a group once even if its members could be listed in several orders.

Switching the meanings of

n

and

r

.

Correction: Use

n

for the total available and

r

for the number selected.

Using a combination when selected people receive different roles.

Correction: Check whether switching who receives a role changes the result. A combination applies when the chosen items have no distinct roles or positions.

Giving only a calculator result.

Correction: Show the combination setup and state what the final number counts.

Lesson summary

A combination counts an unordered selection.
For $n$ available items and $r$ chosen items, use $\binom{n}{r}=\frac{n!}{r!(n-r)!}$ .
Use combinations when changing the listing order does not create a different result.
Choosing $r$ items is equivalent to choosing the $n-r$ items left out.

Check your understanding

Question 1

A club chooses 2 members from 7 to attend a meeting. The two attendees have no different roles. How many pairs can be chosen?

14
21
42
49

Show answer and explanation

21

The pair is unordered, so use a combination:

\binom{7}{2}=\frac{7\times6}{2}=21

.

Question 2

Which situation is an unordered selection?

Choosing a captain and a deputy from a team
Choosing 3 students for a group with no assigned roles
Assigning first, second, and third place
Choosing a speaker and a note-taker

Show answer and explanation

Choosing 3 students for a group with no assigned roles

The three students have no distinct roles, so changing their listing order does not make a new group.

Question 3

How many ways can 8 items be selected from 10 when order does not matter?

20
45
80
90

Show answer and explanation

45

Choose the 2 items left out instead:

\binom{10}{8}=\binom{10}{2}=45

.

Key terms

Selection: A choice of items from a larger collection.
Unordered: A selection where the order or listing of the chosen items does not change the result.
Combination: An unordered selection of a specified number of items from a larger group.
Factorial: The product of a positive whole number and all positive whole numbers below it; for example, $5!=5\times4\times3\times2\times1$ .

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation P6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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P6 · Count unordered selections using combinations

Alberta Grade 12 Mathematics

Count groups when the order of the selected items does not matter

What you will learn

1. Review: what makes a different result?

2. Combination notation and the rule

3. A small example and a useful relationship

4. Set up and interpret a combination

Choose a volunteer team

Select books for a display

Form a planning group by choosing who is left out

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-2

About this lesson and its review