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Math Notes

Counting Methods, With Repetition, Order Matters: Fundamental Counting Principle

Introduction

The Fundamental Counting Principle described on this page begins with a basic property of counting.  If we can break the problem up into smaller parts and count these smaller parts more easily, then we can multiply the number of the smaller parts together to discover the number of the whole.

The simplest example is with a two-digit number.  One possible method of counting the number of two digit numbers would be to make a list of them all and then count them.  But this is tedious and time consuming.  Instead, what we can do is break up our two-digit number into two parts: the first digit and the second digit.  For the sake of keeping track of these parts we can represent them as spaces or as boxes to hold their places.

_______      ________
digit 1          digit 2

Now what we need to determine is the number of possible values for each part.  How many numbers can we put into the first digit?  Well, we didn't specify that we are allowed to use numbers starting with zero so there is a little bit of ambiguity in the problem, but let's assume that we mean numbers that we would actually write as two digits normally.  That means a number starting with 0 isn't possible, so we are left with 1, 2, 3, 4, 5, 6, 7, 8 and 9 to fill the first digit.  That's nine possibilities.

Now, how many numbers can we put in the second digit?  Well, now we can use 0 because we do write numbers that end in zero as two digits.  So our options for the second digit are all ten numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.  So now our places look like this:

___9___       ___10___
digit 1             digit 2

To get the number of two digit numbers, we just multiply 9*10 = 90.  So there are 90 two-digit numbers.  We know from experience that the smallest one is 10 and the biggest one is 99.

We can extend this principle to much more complicated situations, and I will refer back to it from time to time.

Worked Examples

As the title of this page suggests, the Fundamental Counting Principle is most often used whenever Order Matters.  It can be used for cases that either allow or disallow Repetition, but I will concentrate my examples here on cases With Repetition, because the cases with No Repetition will be discussed in Counting Methods III.

A. Suppose we wished to determine the number of three-letter words whose middle letter is a vowel and whose first and last letters were consonants.  Right away we see that we have a three-letter word, so it makes sense to try to divide this problem up into three parts.

_______     ________   ________
letter-1        letter-2         letter-3

Let's start with the middle letter.  How many vowels are there in the alphabet?  Well, there's a, e, i, o and u.  If we were talking about real English words, we might also wish to count y; however, most problems of this type consider y a consonant.  That gives us five possibilities for the middle letter.  The rest of the alphabet is consonants, so since there are 26 total letters, 26-5 = 21 consonants.  (Note: if you choose to do this problem counting y as a vowel, remember that it's also a consonant.)  This gives us the following:

___21___     ___5___   ___21____
letter-1        letter-2         letter-3

The total possible words of this configuration would then be 21*5*21 = 2205.

B. How many possible telephone numbers are there for each telephone exchange?

First, we have to figure out what this problem is asking.  The telephone exchange is the first three digits of a telephone number.  For television shows, the exchange 555 has been set aside just for them--so they never have to use a real number.  Another way of asking this question is: choose one telephone exchange (let's say yours), how many people can have the same first three digits as you?

Since a telephone has seven digits, we could start out this way:

_______     ________     ________   _______    ________    ________      _________

We have seven slots, but since we are talking about fixing the first three digits, the exchange part of the number, we are really starting out with:

____1___     ____1___     ___1____   _______    ________    ________      _________

(Suppose I choose 555 as the exchange I'm working with, then there is only one possible choice for the first digit, and only one possible choice for the second, and one possible choice for the third digit.  The other digits are what we still have to determine.  This will be true for any fixed exchange.)

The remaining four slots can be filled in with any number.  There are no restrictions on the last four digits of a phone number.  Phone companies like to give certain numbers to businesses and certain others to residents but that doesn't affect the total number of numbers.  Each remaining digit can be any of: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.  So, we'll fill in the remaining slots with 10's.

____1___     ____1___     ___1____   ___10___    ___10____    ___10____      ___10____

So the answer to our question is 1*1*1*10*10*10*10 = 10,000.

C. How many alphanumeric 6-letter passwords are there?

By alphanumeric, we mean any letter or number.  For each letter or space in the password we have ten number possibilities and 26 letter possibilities.  Assuming the simplest scenario, where case doesn't matter and we aren't allowed to use special characters like # or what have you, each space in the password has 36 possible values.  This gives us:

___36___    ___36___   ___36___   ___36___   ___36___   ___36___

The answer to our problem then is 36*36*36*36*36*36 or (36)⁶ =   2,176,782,336 possible combinations.

Problem Solving Tips

What I try to do when solving these kinds of problems is look for a few key things before I begin, and as I go through the problem. 

• Always, first read the problem all the way through.  You want to find the question in the problem; that will tell you what exactly you're looking for. 
• You want to find out how many parts you can divide the problem into.  Sometimes the problem will be explicit, like examples A and C where we had a three-digit word or a six-letter password.  Sometimes the problem will be less explicit and expect you to know something common about the word, as in example B where we needed to know that telephone numbers are seven digits and the first digit is the exchange.  For younger students, this might not seem so obvious because so many cities are going to ten-digit phone numbers. 
• You also want to look for any restrictions on any of the digits.  In example B we saw that we wanted to know only about each exchange or a single exchange, so we were fixing the first three digits of the number to a single possibility.  In example A, we restricted the middle letter to just vowels and the outside letters to just consonants.  Even though there are 26 possible letters, we aren't allowed to use all of them in each position.  Other examples of this that we didn't use might be even/odd numbers, numbers less than a certain number, etc.
• Since you can also use this method for permutations, you also want to check to see whether or not repetition is allowed.  (Am I allowed to use the same digit or letter twice?  For instance is 99 or bib allowed?  Check out Counting Methods III to see examples of when it isn't allowed.   With letters and numbers repetition is usually fine unless the problem specifies otherwise.)
• You also want to look for any possible ambiguities and attempt to resolve these logically.  The problem may lend itself to one interpretation over another because of the context.  Problems which are worded properly will disambiguate, but we can't always rely on this in the real world.  You may need to get additional information (if this is possible).  If this isn't possible, chose the interpretation closest to everyday experience.

Additional Problems with Answers

1. How many five digit numbers are there?

2. How many five digit even numbers are there?

3. In Ohio, the regular license plates have three letters, followed by four numbers.  How many different license plates are there?

4. Suppose you flip a coin five times?  How many different sets of five flips are there?

5. Suppose you are going to a restaurant that serves items a la carte.  You decide to order three things: one meat dish, one vegetable dish and one bread.  You could order beef, chicken or fish for the meat dish.  You could order carrots, beans, broccoli, or corn for the vegetable dish.  And you could order biscuits, muffins, or sweet bread for the bread dish.  How many different possible dinners could be served?

6. Suppose you are dealing four cards from a deck and you end up with one card from each suit.  How many different hands of this kind are there?  (Hint: remember that there are 13 cards in each suit.)

7. A couple is interested in having children.  They've decided on having three.  Suppose they've chosen the names Chris, Kelly and Pat (all unisex names).  How many possible ways can they name their three children? (They won't give two children the same name.)

8. How many radio station call signs are there?  (Hint: They have to start with W or K and have four letters.)

9. How many ways are there to roll three six-sided dice?  (Pretend you are rolling one die three times.)

10. Suppose you have a keypad with twelve keys on it for your security system.  How many different four-key passcodes are there?

Challenge Problems:

11. Suppose you have four friends and you want to buy tickets to the midnight showing of Star Wars: Episode III.  How many different ways can the five of you wait in line? (Hint: Once you pick one friend to stand first in line, they can't stand anywhere else in line.  If there are five ways for the first person to stand in line, how many people are left to stand second? third? etc.?)

12. How many 8-digit passwords are there using alphanumeric characters and case matters?  You may want to state your answer in scientific notation.

13. If it takes you five seconds to check each of the passwords in problem #12, how long will it take to test them all?  State your answer in years.

Answers to Problems:

1. 10⁵ = 100,000, if you allow 0 as a first digit, otherwise 9*10⁴=90,000
2. 10⁴*5 = 50,000
3. 26³*10⁴ = 175,760,000
4. 2⁵ = 32
5. 3*4*3 = 36
6. 13⁴ = 28,561
7. 3*2*1 = 6
8. 2*26³ = 35,152
9. 6³ = 216
10. 12⁴ = 20,736
11. 5*4*3*2*1 = 120
12. 62⁸ = 2.18 x 10¹⁴
13. Divide your answer in 12 by (60*60*24*365) = 6,923,519 years

Links to Outside Sources

College Algebra Tutorial on The Fundamental Counting Principle
Objective A1 Fundamental Counting Principle
Problem Bank - Fundamental Counting Principle
11.1 Fundamental Counting Principle
 

Links to Supporting Topics

Counting Methods II -- With Repetition, Order Doesn't Matter
Counting Methods III -- No Repetition, Order Matters: Permutations
Counting Methods IV -- No Repetition, Order Doesn't Matter: Combinations
7.6 - Counting Principles

History

If you know of any links the discuss the history, origin or development of the Fundamental Counting Principle, please email me at the address below.