Section 5.2: The Fundamental Counting Principle

Last updated
Save as PDF

Page ID: 215604

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Learning Objectives

Use the fundamental counting principle

The Fundamental Counting Principle (also called the Multiplication Principle) is a counting method that helps us determine the total number of ways multiple events can occur.

For example, you want to buy an 1-scoop ice cream cone on a hot summer day. The ice cream shop offers 3 flavors (Vanilla, Chocolate, and Strawberry), and 2 cone types (Waffle cone and Sugar cone). You would like to know how many different 1-scoop ice cream cone combinations can be made.

Here is what we know:

3 flavors: Vanilla, Chocolate, and Strawberry
2 cone types: Waffle cone and Sugar cone
Two independent events: choosing flavor + choosing cone type

Thus, each of the 3 flavors can be paired with any of the 2 cone types as follows:

Vanilla gives you: 2 combinations (Vanilla + Waffle cone; Vanilla + Sugar cone)
Chocolate gives you: 2 combinations (Chocolate + Waffle cone; Chocolate + Sugar cone)
Strawberry gives you: 2 combinations (Strawberry + Waffle cone; Strawberry + Sugar cone)

Therefore, there are 2 + 2 + 2 = 6 combinations. This is exactly the same as: 3 $\cdot$ 2 = 6. Extending this idea leads us to the Fundamental Counting Principle.

Definition: The Fundamental Counting Principle

If one event can occur in m ways and a second event can occur in n ways, then both events together can occur in m $\cdot$ n ways.

This rule can extend to any number of events:

If there are k events that can occur in n₁, n₂, n₃, ..... ,n_k ways respectively, then all events together can occur in n₁ $\cdot$ n₂ $\cdot$ n₃ $\cdot$ ... $\cdot$ n_kways.

Note: In the 1-scoop ice cream cone example, it is possible to generate the answer of 6 different ways through several different methods in addition to the Fundamental Counting Principle. We could have used a systematic table listing all flavor-cone pairs, or drawn a tree diagram showing all possible branches from each choice. Tree diagrams will be discussed later in Section 5.4.

However, as we consider problems with more events and more possibilities, these alternative methods become increasingly cumbersome and time-consuming. For instance, imagine if the ice cream shop offered 10 flavors, 4 cone types, 5 toppings, and 3 sizes—that would result in 10 $\cdot$ 4 $\cdot$ 5 $\cdot$ 3 = 600 combinations! Creating a table with 600 entries or drawing a tree diagram with 600 branches would be impractical and prone to errors.

The Fundamental Counting Principle provides a quicker, easier, and more efficient method that scales beautifully with complexity. Rather than laboriously listing every possibility, we can simply identify the number of choices for each independent event and multiply them together. This makes the Fundamental Counting Principle not just a mathematical concept, but an essential problem-solving tool for tackling real-world counting scenarios efficiently and accurately.

Example #5.2.1 🤔

You are dressing for a job interview. You open your closet and see 4 dress shirts and 3 dress pants. How many different outfits can you make?

(Note: Assume that all shirts are distinct from one another and all pants are distinct from one another, unless otherwise noted. Apply this assumption for all similar problems in this section.)

✅ Solution:

Using the Fundamental Counting Principle, there are 4 different shirts and 3 different pants.

Therefore, there are 4 $\cdot$ 3 = 12 different outfits that can be made.

Example #5.2.2 🤔

A new restaurant has opened with a menu offering a choice of 3 salads, 8 main dishes, and 7 desserts. How many different meals consisting of one salad, one main dish, and one dessert are possible?

✅ Solution:

Using the Fundamental Counting Principle, there are 3 salads, 8 main dishes, and 7 desserts.

Therefore, there are 3 $\cdot$ 8 $\cdot$ 7 = 168 different meals that can be made.

Example #5.2.3 🤔

A particular video game allows users to create their own avatar (character). There are 8 skin-tone colors, 12 hair styles, 6 eye colors, 15 outfit options, and 5 accessories. How many different avatars can be created if exactly one option from each category must be selected?

✅ Solution:

Using the Fundamental Counting Principle, there are 8 skin-tone colors choices, 12 hair style choices, 6 eye color choices 15 outfit option choices, 5 accessory choices.

Therefore, there are 8 $\cdot$ 12 $\cdot$ 6 $\cdot$ 15 $\cdot$ 5 = 43,200 different video game characters that can be made.

The three examples presented above represent basic applications of the Fundamental Counting Principle where all choices are independent and unrestricted. As we progress to more advanced applications, we encounter problems that incorporate constraints, limitations, and specific selection criteria.

These enhanced scenarios better reflect real-world situations where complete freedom of choice is rare. In the problems that follow, you will learn to navigate restrictions such as prohibited first digits, mandatory selections from distinct categories, and varying numbers of available options. The fundamental approach remains unchanged:

Step 1: Determine how many events need to occur;

Step 2: Determine the number of possibilities per event;

Step 3: Use the Fundamental Counting Principle (multiply through) to determine the total number of outcomes.

If you are having trouble with Step 1, it may help to draw an empty slot for each event, as shown in Example #5.2.4 below.

Example #5.2.4 🤔

From 1980 to present, California has used a standard 7-character license plate format for passenger vehicles: N-LLL-NNN (1 number, 3 letters, 3 numbers). An example is shown below to the right. Assuming no restrictions apply and characters may be repeated, how many different license plates are possible?
Note: The digits 0-9 are available for the number positions (10 choices each) and the letters A-Z are available for the letter positions (26 choices).

✅ Solution:

So, the best strategy here is to start out with a blank license plate. For all intent purposes, we will have seven empty "slots".

For the first slot, we need a number. There are 10 different letter to choose from, so

For the next three slots, we need a letter in each slot. There are 26 different numbers to choose from, so

10 26 26 26

For the last three slots, we need a number in each slot. There are 10 different letters to choose from, so

10 26 26 26 10 10 10

Using the Fundamental Counting Principle, we now multiply all those choices in each of their respective slots.

Therefore, there are 10 $\cdot$ 26 $\cdot$ 26 $\cdot$ 26 $\cdot$ 10 $\cdot$ 10 $\cdot$ 10 = 175,760,000 seven-digit phone numbers that can be made.

Example #5.2.5 🤔

You received a new ATM card in the mail. You need to select a passcode consisting of exactly 4 digits, where each digit can be chosen from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Assuming that digits can be repeated (for example, 1111 or 2002 are valid passcodes), how many different passcodes are possible?

✅ Solution:

Using the Fundamental Counting Principle, there are 10 different digits for the first slot, 10 different digits for the second slot, 10 different digits for the third slot, and 10 different digits for the fourth slot.

Therefore, there are 10 $\cdot$ 10 $\cdot$ 10 $\cdot$ 10 = 10,000 passcodes that can be made.

Example #5.2.6 🤔

You received a new ATM card in the mail. You need to select a passcode consisting of exactly 4 digits, where each digit can be chosen from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Assuming that digits cannot be repeated (for example, 1234 or 5792 are valid passcodes, while 1123 or 3455 are not valid passcodes), how many different passcodes are possible?

✅ Solution:

Using the Fundamental Counting Principle, there are 10 different digits for the first slot, 9 different digits for the second slot, 8 different digits for the third slot, and 7 different digits for the fourth slot.

Therefore, there are 10 $\cdot$ 9 $\cdot$ 8 $\cdot$ 7 = 5,040 passcodes that can be made.

Example #5.2.7 🤔

A phone number consists of 7 digits, where each digit can be chosen from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. However, the first digit cannot be 0 or 1. If repetition of digits is allowed, how many different 7-digit phone numbers are possible?

✅ Solution:

Using the Fundamental Counting Principle, there are 8 different digits for the first slot and 10 different digits for each of the second through seventh slots.

Therefore, there are 8 $\cdot$ 10 $\cdot$ 10 $\cdot$ 10 $\cdot$ 10 $\cdot$ 10 $\cdot$ 10 = 8,000,000 seven-digit phone numbers that can be made.

Example #5.2.8 🤔

A test is divided into two parts. The first part has 6 multiple-choice questions where each question can be answered as A, B, C, or D. The second part has 4 True/False questions, where each question can be answered as either True or False. How many different answer keys could be created for this entire test?

✅ Solution:

Using the Fundamental Counting Principle, there are 4 choices for each of the 6 multiple-choice questions and 2 choices for each of the 4 true/false questions.

Therefore, there are 4 $\cdot$ 4 $\cdot$ 4 $\cdot$ 4 $\cdot$ 4 $\cdot$ 4 $\cdot$ 2 $\cdot$ 2 $\cdot$ 2 $\cdot$ 2 = 4⁶ $\cdot$ 2⁴ = 4,096 $\cdot$ 16 = 65,536 different ways to make an answer key.

Section 5.2: The Fundmental Counting Principle [In-Class Exercises]

A laptop can be configured with one of each of the following to choose from: 3 processor types, 2 RAM sizes, 3 storage sizes, 2 graphics options, and 4 color options. How many different laptops can be made?

When ordering online from Yogurtland, they usually offer 12 different flavors of yogurt and 24 different toppings. How many different orders could you make from one flavor and one topping?

A quiz has 6 true/false questions. How many different ways can a student answer all 6 questions?

Arizona's standard passenger license plates feature a six-character "alphabet soup" format. The six characters can be any number 0-9 or any letter from A-Z, with the exceptions of I, O, Q, and U to avoid confusion. How many different six-character license plates can be made?

A social security number contains 9 digits using any digit from 0-9 inclusive. How many different social security numbers can be made?

An ATM at a particular bank requires a 6-digit numeric pin. How many 6-digit numeric pins can be made?

An ATM at a particular bank requires a 6-digit numeric pin. How many 6-digit numeric pins can be made, if no digits can be repeated?

At BJ's Restaurant & Brewhouse, you can customize your pizza with the following toppings.

Meat: Cupping Pepperoni, Italian Sausage, Smoked Bacon, Chicken, Smoked Ham, Housemade Meatballs, & Anchovies
Veggie: Mushrooms, Black Olives, Jalapenos, Pineapple, Green Bell Peppers, Fresh Basil, Roasted Garlic, Fire Roasted Red Peppers, Onions, Red Onions, & Green Onions

If a pizza was ordered with a choive of one meat topping and one veggie topping, how many different ways can you order a pizza?

As of 2026, in the NFL, there have been 60 Super Bowls that have been played. One of sixteen NFC (National Football Conference) teams always plays one of the AFC (American Football Conference) teams in the Super Bowls. There have been several NFL team pairings that have faced each other in the Super Bowl more than once. In fact, it has actually happened 9 different times with same pairings. They are:

Dallas Cowboys vs. Pittsburgh Steelers in Super Bowls 10, 13, & 30
Miami Dolphins vs. Washington Redskins in Super Bowls 7 & 17
Cincinnati Bengals vs. San Francisco 49ers in Super Bowls 16 & 23
Buffalo Bills vs. Dallas Cowboys in Super Bowls 27 and 28
New England Patriots vs. New York Giants in Super Bowls 42 and 46
New England Patriots vs. Philadelphia Eagles in Super Bowls 39 & 52
New England Patriots vs. St. Louis/Los Angeles Rams in Super Bowls 36 and 53
Kansas City Chiefs vs. San Francisco 49ers in Super Bowls 54 and 58
New England Patriots vs. Seattle Seahawks in Super Bowls 49 and 60

How many different pairings could there be for any NFC team to play an AFC team?

California State Department of Motor Vehicles offers personalized “California Museums Snoopy” license plates and can have up to 6 custom characters of numbers and/or letters. How many different 6-character license plates are possible if there are no restrictions?

$Homework 5.3.10.png$

Answers

$3 \cdot 2 \cdot 3 \cdot 2 \cdot 4 = 144$ different laptops
$12 \cdot 24 = 288$ different orders
$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^6 = 64$ different orders
$32 \cdot 32 \cdot 32 \cdot 32 \cdot 32 \cdot 32 = 32^6 = 1,073,741,824$ different license plates
$10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 10^9 = 1,000,000,000$ different social security numbers
$10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 10^6 = 1,000,000$ different pins
$10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 = 10^6 = 151,200$ different pins
$7 \cdot 11 = 77$ different pizzas
$16 \cdot 16 = 256$ different pairings
$36 \cdot 36 \cdot 36 \cdot 36 \cdot 36 \cdot 36 = 36^6 = \text{2,176,782,336}$ different license plates

Search

Text Color

Text Size

Margin Size

Font Type

✅ Solution:

✅ Solution:

✅ Solution:

✅ Solution:

✅ Solution:

✅ Solution:

✅ Solution:

✅ Solution: