Math Portal
Introductory Statistics
Section 2.1 - The Fundamental Counting Principle
The Fundamental Counting Principle states that if a first task can be done in any one of m different ways and, after that task is completed, a second task can be done in any one of n different ways, then both tasks can be done, in the order given, in m times n different ways.
EXAMPLE: How many different meals consisting of one sandwich and one drink are possible from a selection of 10 different sandwiches and 6 different drinks?
SOLUTION: This problem consists of two tasks: selecting a sandwich and then selecting a drink. You have 10 ways to select a sandwich; following that selection, you have 6 ways to select a drink. Therefore, you have 10 ∙ 6 = 60 different meals that consist of one sandwich and one drink.
This counting principle can be extended to any number of tasks. Thus, in general, for a sequence of k tasks, if a first task can be done in any one of n(1) different ways and, after that task is completed, a second task can be done in any one of n(2) different ways, and, after the first two tasks have been completed, a third task can be done in any one of n(3) different ways, and so on to the k-th task, which can be done in any one of n(k) different ways, then the total number of different ways the sequence of k tasks can be done is n(1) + n(2) + n(3) + ... + n(k).
In other words, simply add together number of possibilities for each of the different tasks.
EXAMPLE: A code for a home alarm system consists of four digits that must be entered in a specific order. Each of the digits zero through nine may be used in the code, and repetition of digits is allowed. How many different four-digit codes are possible?
SOLUTION: This problem consists of four tasks. You have 10 ways to select the first digit, 10 ways to select the second digit, 10 ways to select the third digit, and 10 ways to select the fourth digit. Therefore, you have 10∙10∙10∙10 = 10,000 different possible codes.
EXAMPLE: There are 20 separate candidates for three vice-president (VP) positions at a university. Assuming all 20 candidates are qualified to be selected for any one of the three VP positions, how many different ways can the positions be filled?
SOLUTION: This problem consists of three tasks. You have 20 candidates available to fill the first VP position; after filling that position, you have 19 candidates remaining that are available to fill the second VP position, and after filling the first and second VP positions, you have 18 candidates remaining to fill the third VP position. Therefore, you can fill the three VP positions in 20∙19∙18=6840 different ways.
Note that this counting technique yields results in which the order of the elements determines different outcomes. For example, the codes 2413 and 3124 for a home alarm system consist of the same digits, but are considered different codes because the order in which the digits appear is different.
The Fundamental Counting Principle states that if a first task can be done in any one of m different ways and, after that task is completed, a second task can be done in any one of n different ways, then both tasks can be done, in the order given, in m times n different ways.
EXAMPLE: How many different meals consisting of one sandwich and one drink are possible from a selection of 10 different sandwiches and 6 different drinks?
SOLUTION: This problem consists of two tasks: selecting a sandwich and then selecting a drink. You have 10 ways to select a sandwich; following that selection, you have 6 ways to select a drink. Therefore, you have 10 ∙ 6 = 60 different meals that consist of one sandwich and one drink.
This counting principle can be extended to any number of tasks. Thus, in general, for a sequence of k tasks, if a first task can be done in any one of n(1) different ways and, after that task is completed, a second task can be done in any one of n(2) different ways, and, after the first two tasks have been completed, a third task can be done in any one of n(3) different ways, and so on to the k-th task, which can be done in any one of n(k) different ways, then the total number of different ways the sequence of k tasks can be done is n(1) + n(2) + n(3) + ... + n(k).
In other words, simply add together number of possibilities for each of the different tasks.
EXAMPLE: A code for a home alarm system consists of four digits that must be entered in a specific order. Each of the digits zero through nine may be used in the code, and repetition of digits is allowed. How many different four-digit codes are possible?
SOLUTION: This problem consists of four tasks. You have 10 ways to select the first digit, 10 ways to select the second digit, 10 ways to select the third digit, and 10 ways to select the fourth digit. Therefore, you have 10∙10∙10∙10 = 10,000 different possible codes.
EXAMPLE: There are 20 separate candidates for three vice-president (VP) positions at a university. Assuming all 20 candidates are qualified to be selected for any one of the three VP positions, how many different ways can the positions be filled?
SOLUTION: This problem consists of three tasks. You have 20 candidates available to fill the first VP position; after filling that position, you have 19 candidates remaining that are available to fill the second VP position, and after filling the first and second VP positions, you have 18 candidates remaining to fill the third VP position. Therefore, you can fill the three VP positions in 20∙19∙18=6840 different ways.
Note that this counting technique yields results in which the order of the elements determines different outcomes. For example, the codes 2413 and 3124 for a home alarm system consist of the same digits, but are considered different codes because the order in which the digits appear is different.