Principles of Counting

Factorial Notation

The Fundamental Counting Principle

The Counting Principle forAlternative Cases

Permutation

Combination

Factorial NotationFor any positive integer n, n! means:

n (n – 1) (n – 2) . . . (3) (2) (1)

0! will be defined as equal to one

Examples:4! = 4•3 •2 •1 = 24

The factorial symbol only affects the number it follows unless grouping symbols are used.

3 •5! = 3 •5 •4 •3 •2 •1 = 360

( 3 •5 )! = 15! = big number

The Fundamental Counting Principle

If activity 1 can be done in n1 ways, activity 2 can be done in n2 ways, activity 3 can be done in n3 ways, and so forth; then the number of ways of doing these activities on a specified order is the product of n1, n2, n3 and so forth. In symbols, nnnnn 321

Example 1:

Suppose a school has three gates, in how many ways can a student enter and leave the school?

Example 2:

In a medical study, patients are classified according to whether they have blood type A, B, AB or O, and also according to whether their blood pressure is low, normal, or high. In how many different ways can a patient thus be classified.

Example 3:

A new car dealer offers a car in four body styles, in ten colors, and with a choice of three engines. In how many ways can a person order one of the cars?

Example 4:

A test consists of 15 multiple choice questions, with each question having four possible answers. In how many different ways can a student check off one answer to each question?

Example 5:

How many different 4-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 if: (a) repetition is not allowed? How many of these numbers are even? How many are these numbers are odd? (b) repetition is allowed?

The Counting Principle of Alternative Cases

Suppose the ways of doing an activity can be broken down into several alternative cases where each case does not have anything in common with the other cases. If case 1 can be done in n1 ways, case 2 can be done in n2 ways, case 3 can be done in n3 ways, and so on, then the number of ways the activity can be done is the sum of n1, n2, n3 and so on cases. In symbols, nnnnn 321

Permutation

The term permutation refers to the arrangement of objects with reference to order or it may be defined as an arrangement of all or part of a set of objects.

Linear Permutation

The number of permutations of n distinct objects taken all together is n!.

Example:How many different signals can be made using five flags if all flags must be used in each signal?

Permutation of n Elements Taken r at a TimeThe arrangement of n objects in a specific order using r objects at a time is given by

where r < nExample:

Suppose there are eight machines, but only three spaces in the display room available for the machines. In how many different ways can the 8 machines be arranged in the three available spaces

!!

rn

nP rn

Circular Permutation

The arrangement of n objects in a circular pattern is given by the formula

Example:In how many ways can six persons be seated around a circular table?

!1 nP

Permutation of Things Not All DifferentThe number of distinct permutations of n objects of which r1 are alike, r2 are alike, r3 are alike, … etc. is

Example:How many different permutations can be made from the letters of the word “STATISTICS”?

!!!!

!

321 nrrrr

nP

CombinationSuppose we are interested only in the number of different ways that r objects can be selected from a given number of objects. If the order of the objects is not important, the total number of orders or arrangement is called combination. The number of combinations of n objects taken r at a time is denoted by nCr and is given by the formula: !!

!

rrn

nCrn

Example 1:

In order to survey the opinions of costumers at local malls, a researcher decides to select 5 malls from a certain area with a total of 9 malls. How many different ways can the selection be made?

Example 2:

The general manager of a fast-food restaurant chain must select 6 restaurants from 10 for a promotional program. How many different possible ways can this selection be done?

Problem 1:

In how many ways can 5 people line up for a group picture if (a) two want to stand next to each other? (b) two refuse to stand next to each other?

Problem 2:

In how many ways can 8 beads be put together to form a round bracelet?

Problem 3:

A committee of 5 people must be selected from 5 accountants and 8 educators. How many ways can the selection be done if there are 3 educators in the committee?

Problem 4:

In a club there are 8 women and 5 men. A committee of 4 women and 2 men is to be chosen. How many possibilities are there?

Problem 5:

A committee of 5 people must be selected from 5 accountants and 8 educators. How many ways can the selection be done if there are at least 3 educators in the committee?

Problem 6:

How many different triangles can be formed using the vertices of an octagon? Pentagon? Hexagon?