2. Basic Principles of Counting

Counting

An efficient way of counting is necessary to handle large masses of statistical data (e.g. the level of inventory at the end of a given month, or the number of production runs on a given machine in a 24 hour period, etc.), and for an understanding of probability.

On this page...

Number of outcomes
Addition rule
Multiplication rule

In this section, we shall develop a few counting techniques. Such techniques will enable us to count the following, without having to list all of the items:

the number of ways,
the number of samples, or
the number of outcomes.

Before we learn some of the basic principles of counting, let's see some of the notation we'll need.

Number of Outcomes of an Event

As an example, we may have an event E defined as

E = "day of the week"

We write the "number of outcomes of event E" as n(E).

So in the example,

n(E) = 7,

since there are 7 days in the week.

Addition Rule

Let E₁ and E₂ be mutually exclusive events (i.e. there are no common outcomes).

Let event E describe the situation where either event E₁ or event E₂ will occur.

The number of times event E will occur can be given by the expression:

n(E) = n(E₁) + n(E₂)

Tip

In counting and probability, "OR" usually requires us to ADD.

where

n(E) = Number of outcomes of event E

n(E₁) = Number of outcomes of event E₁

n(E₂) = Number of outcomes of event E₂

[We see more on mutually exclusive events later in this chapter.]

Example 1

Consider a set of numbers S = {-4, -2, 1, 3, 5, 6, 7, 8, 9, 10}

Let the events E₁, E₂ and E₃ be defined as:

E = choosing a negative or an odd number from S;

E₁= choosing a negative number from S;

E₂ = choosing an odd number from S.

Find n(E).

Answer

Example 2

In how many ways can a number be chosen from 1 to 22 such that

(a) it is a multiple of 3 or 8? (b) it is a multiple of 2 or 3?

Answer

Multiplication Rule

Now consider the case when two events E₁ and E₂ are to be performed and the events E₁ and E₂ are independent events i.e. one does not affect the other's outcome.

[We see more on independent events later in this chapter.]

Example

Say the only clean clothes you've got are 2 t-shirts and 4 pairs of jeans. How many different combinations can you choose?

Answer

We can think of it as follows:

jeans

We have 2 t-shirts and with each t-shirt we could pick 4 pairs of jeans. Altogether there are

2 × 4 = 8 possible combinations.

We could write

E₁ = "choose t-shirt" and

E₂ = "choose jeans"

Multiplication Rule in General

Suppose that event E₁ can result in any one of n(E₁) possible outcomes; and for each outcome of the event E₁, there are n(E₂) possible outcomes of event E₂.

Together there will be n(E₁) × n(E₂) possible outcomes of the two events.

Tip

In counting and probability, "AND" usually requires us to MULTIPLY.

That is, if event E is the event that both E₁ and E₂ must occur, then

n(E) = n(E₁) × n(E₂)

In our example above,

n(E₁) = 2 (since we had 2 t-shirts)

n(E₂) = 4 (since there were 4 pairs of jeans)

So total number of possible outcomes is given by:

n(E) = n(E₁) × n(E₂) = 2 × 4 = 8

Example 3

What is the total number of possible outcomes when a pair of coins is tossed?

Answer

Example 4

The life insurance policies of an insurance company are classified by:

age of the insured:
- under 25 years,
- between 25 years and 50 years,
- over 50 years old;
sex;
marital status:
- single or
- married.

What is the total number of classifications?

Answer

Example 5

For our clothes problem above, say we found 3 caps that we could wear with our 2 t-shirts and 4 pairs of jeans. How many different combinations could we choose from now?

Answer

Example 6

Image source

In the excellent Monoface (external site), we can change the head, left and right eyes, nose and mouth of some zany guys who work at momo-1.com.

We are told that there are 759,375 possible faces. Where does this number come from?

If they were to let us change the chin as well, how many possible combinations would there be?

Answer

1. Factorial Notation

3. Permutations

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