High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Request a demo

In order to access this I need to be confident with:

Multiplication Factors and multiples Greatest common factor Least common multiple Probability Independent events Mutually exclusive events

Math resources Statistics and probability Compound prob.

Fundamental C P

Fundamental counting principle

Here you will learn about the fundamental counting principle, including selecting from one set and selecting from multiple sets.

Students will first learn about the fundamental counting principle as part of statistics and probability in $7$ th grade.

What is the fundamental counting principle?

The fundamental counting principle is a method for finding the total number of ways of selecting items from a set or sets.

The fundamental counting principle (also called the fundamental principle of counting) states that if one event can occur in $m$ ways and a second independent event can occur in $n$ ways, then the total number of ways both events can occur in sequence is $m \times n.$

In other words, if there are multiple independent events, the total number of possible outcomes is found by multiplying the number of ways each event can occur.

For example,

A teacher needs to select one boy and one girl from his class to show a visitor around the school. There are $4$ boys and $3$ girls in his class.

The number of different pairs of children can be calculated by thinking about how many different options there are for each boy.

Each boy could be paired with $3$ girls.

As there are $4$ boys, this means there will be $4$ groups of $3$ pairings.

$4\times 3=12,$ so there are $12$ different pairs (permutations) the teacher could select.

If the boys are called $A, \, B, \, C$ and $D$ and the girls are called $X, \, Y, \, Z$ the sample space would be:

If the teacher decided to just select two girls from his class, you would need to think more carefully.

There would be $3$ options for the first choice, and then $2$ options for the second choice. However, calculating $3\times 2$ would give double the amount of options as it would include each pair of girls twice, and the order of selecting the girls doesn’t matter.

So there are $\cfrac{3\times 2}{2}=3$ pairs of girls to select.

The possible combinations would be: $XY \, ($ same as $YX), \, XZ \, ($ same as $ZX)$ and $YZ \, ($ same as $ZY).$

What is the fundamental counting principle?

Common Core State Standards

How does this relate to $7$ th grade math?

Grade 7 – Statistics & Probability (7.SP.C.8)
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

[FREE] Probability Worksheet (Grade 7 to 12)

Use this quiz to check your grade 7 to 12 students’ understanding of finding probability. 15+ questions with answers covering a range of 7th to 12th grade probability topics to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Probability Worksheet (Grade 7 to 12)

DOWNLOAD FREE

How to use the fundamental counting principle

In order to use the fundamental counting principle:

Identify the number of sets to be selected from.
Identify the number of items to select from each set.
Multiply the number of items in each set. If selecting two items from a set, calculate $\textbf{n}\times \left( \textbf{n}-\bf{1} \right)$ or $\cfrac{\textbf{n}\times \left( \textbf{n}-\bf{1} \right)}{\bf{2}}$ if order doesn’t matter.

Fundamental counting principle examples

Example 1: selecting a pair from two different sets

Arthur has been told he can select a bag of chips and a drink as part of a meal deal.

There are $7$ different flavors of chips and $11$ different drinks. How many possible outcomes could Arthur select?

Identify the number of sets to be selected from.

There are two sets to select from, chips and drinks.

2Identify the number of items to select from each set.

There are $7$ different chip flavors and $11$ different drinks.

3Multiply the number of items in each set. If selecting two items from a set, calculate $\textbf{n}\times \left( \textbf{n}-\bf{1} \right)$ or $\cfrac{\textbf{n}\times \left( \textbf{n}-\bf{1} \right)}{\bf{2}}$ if order doesn’t matter.

7\times 11=77

There are $77$ total choices that Arthur could select.

Example 2: selecting from multiple sets

Bella is clothes shopping and can select a T-shirt, a pair of joggers, and a hoodie as part of a mix-and-match offer.

There are $4$ different colors of T-shirts, $5$ colors for the joggers, and $3$ colors for the hoodie.

What is the number of possibilities Bella could select?

Identify the number of sets to be selected from.

There are three sets to select from.

Identify the number of items to select from each set.

The number of items from the three sets are $4, \, 5,$ and $3.$

Multiply the number of items in each set. If selecting two items from a set, calculate $\textbf{n}\times \left( \textbf{n}-\bf{1} \right)$ or $\cfrac{\textbf{n}\times \left( \textbf{n}-\bf{1} \right)}{\bf{2}}$ if order doesn’t matter.

Using the counting principle, you multiply

$4\times 5\times 3=60$

There are $60$ different possibilities that Bella could select.

Example 3: selecting a pair from one set when order matters

A teacher is trying to pick two students to participate in a quiz. They must also decide which student will go first and which will go second in a round of the quiz. If there are $24$ students to choose from, how many different options does the teacher have?

Identify the number of sets to be selected from.

There is one set to select from.

Identify the number of items to select from each set.

There are $24$ items in the set.

Order matters because the teacher needs to select who will go first and who will go second.

$24\times 23=552$

There are $552$ different options that the teacher could select.

Example 4: selecting a pair from one set when order doesn’t matter

$15$ teams in a league must play each other once. How many games in total will be played?

Identify the number of sets to be selected from.

There is one set to select from.

Identify the number of items to select from each set.

There are $15$ items in the set.

Order doesn’t matter because the teams must only play each other once.

$\cfrac{15\times 14}{2}=105$

There will be $105$ games played in total.

Example 5: selecting two pairs from two different sets

A teacher must choose a Head Boy and Head Girl along with a Deputy Head Boy and a Deputy Head Girl. There are $8$ boys and $10$ girls to select from. How many different combinations are possible?

Identify the number of sets to be selected from.

There are two sets to select from.

Identify the number of items to select from each set.

There are $8$ items in the boy set and $10$ in the girl set.

Order matters because there are different positions to fill.

If there are n items in one set and m items in the other set, you must calculate $n\times \left( n-1 \right)\times m\times \left( m-1 \right).$

$8\times 7\times 10\times 9=5040$

There are $5040$ different combinations.

Teaching tips for fundamental counting principle

Create practice problems using relatable examples, such as choosing different outfits with options for shirts, pairs of pants, and pairs of shoes, or ice cream flavors and toppings. This makes it easier for students to visualize the principle.

Draw a tree diagram to show how each choice expands into more options. This makes the multiplication step more tangible and emphasizes why multiplying each option in sequence gives the total number of choices.

Have advanced students analyze scenarios where outcomes involve multiple layers of choices (like creating a password or license plate with specific conditions).

Easy mistakes to make

Forgetting to multiply by one less if selecting from a single set
A common error is to forget that when selecting two items from a set, the second choice is from a set reduced in size by one. You must assume that the first choice is not replaced unless the question specifically states that it is.

Not knowing when order matters or not
Some students find it confusing to know when the order of a pair of items selected from a single set matters. If there are $n$ items in the set and items are labeled $A, \, B, \, C, \, ….$ calculating $n\times \left( n-1 \right)$ will include the pairs $AB$ and $BA.$ If the question was about teams playing each other only once, you would need to divide this value by $2$ to give the correct number of pairings. If, however, the teams needed to play each other twice then $n\times \left( n-1 \right)$ would be correct.

Practice fundamental counting principle questions

77

84

72

66

For each $7$ starters there are $12$ possible main course choices, so calculate $7\times 12=84.$

126

80

18

192

For each $6$ starters there are $8$ possible main course choices and then $4$ dessert choices, so calculate $6\times 8\times 4=192.$

462

484

231

242

Order doesn’t matter so calculate $22\times 21=462$

and then half this answer

462\div 2=231.

90

100

45

50

Order matters so calculate $10\times 9=90.$

1,764

1,260

882

630

There are $6$ choices for head boy and $7$ for head girl.

Then there are $5$ choices for deputy head boy and $6$ for deputy head girl.

6\times 7\times 5\times 6=1260

72

81

116

99

The number of ways would be $9\times x,$ therefore it must be a multiple of $9.$

Fundamental counting principle FAQs

What is the fundamental counting principle?

The fundamental counting principle is a basic rule in combinatorics that states if there are multiple independent events, the total number of possible outcomes is found by multiplying the number of ways each event can occur.

What are some other names for the fundamental counting principle?

The fundamental counting principle is also known by several other names, including the fundamental counting rule, the multiplication principle or multiplication rule of counting, the counting rule, the rule of product, the product principle, the basic counting principle.

How is the fundamental counting principle used in real life?

The fundamental counting principle is used in real life to calculate the total number of possible outcomes in multi-step processes, such as creating secure passwords, determining travel routes, organizing event seating arrangements, and designing unique license plates. It helps simplify complex choices by breaking down each independent decision point.

The next lessons are

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.