Counting Rules
The task of counting can be made easy!
Talking Points
In this lesson, the following will be our main talking points:

Multiplication rule

Combinations rule

Permutations rule
Learning Objectives
After completing this lesson and also going through the relevant section in the
study guide, the following is expected of you:

You should know and be able to use the counting rules.

You should also be able to do the calculations both manually and using a
calculator.
Multiplication Rule

If an operation can be performed in n ways, and another operation can be
performed in m ways, then either of the two operations can be performed in
(m+n) ways.

This rule can be extended to any finite number of operations. That is, if one
operation can be done in n1 ways, a second one in n2 ways, . . . , a kth one in nk
ways, then one of the k operations can be done in n1 + n2 +. . . + nk ways.
Example 1

A basic meal consists of soup, a sandwich and a beverage. If a person having
this meal has 3 choices of soup, 4 choices of sandwiches and a choice of
coffee or tea as a beverage, how many such meals are possible?
Solution

Choosing soup (taken as operation 1) has 3 possibilities.

Choosing a sandwich (taken as operation 2) has 4 possibilities.

Choosing a beverage (taken as operation 3) has 2 possibilities.

Therefore, the number of choices of meal is equal to 3 × 4 × 2 = 24, according
to the multiplication rule.
Example 2

A PIN to be used at an ATM can be formed by selecting 4 digits from the digits
0, 1, 2, . . . , 9 . How many choices of PIN are there if digits may not be
repeated?
Solution

Choosing first digit (taken as operation 1) has 10 choices, second digit (taken
as operation 2) has 9 choices, third digit (taken as operation 3) has 8 choices,
and fourth digit (taken as operation 4) has 7 choices.

Therefore, the number of choices of 4-digit pin is 10 × 9 × 8 × 7 = 5040,
according to the multiplication rule.
Permutations and
Combinations
Note the difference!
Permutations

A permutation is the number of different arrangements of a group of items
where order matters.

The number of permutations of 𝑛 objects taken 𝑟 at a time is calculated using
the formula below (Permutations formula)
𝑛𝑃𝑟 = 𝑃 𝑛, 𝑟 =
𝑛!
𝑛−𝑟 !
,
with symbol 𝑃 meaning Permutations and parameters 𝑛, 𝑟 as defined above.
Combinations

A combination is the number of different selections of a group of items where
order does not matter.

The number of combinations of a group of 𝑛 objects taken 𝑟 at a time is
calculated using the formula below (Combinations formula):
𝑛𝐶𝑟 = 𝐶 𝑛, 𝑟 =
𝑛!
𝑟! 𝑛−𝑟 !
,
with symbol 𝐶 meaning Combinations and parameters 𝑛, 𝑟 as defined above.
Example 1

Four people (A, B, C, D) serve on a board of directors. A chairman and
vice-chairman are to be chosen from the four people. In how many
ways can this be done?

Answer: Number of ways is 12.

As see below is a list of all possible selections of two people (the
chairman and vice-chairman) from four people.
Example 2

Suppose that the two people to be selected from four people (A, B, C, D) are to
serve a specified similar role. In how many ways can this be done?

Answer: Number of ways = 6.

As seen below is a list of all possible selections of two people from four people
People chosen A and B A and C A and D B and C B and D C and D
Application of formulae

In example 1, the Permutations formula applies with 𝑛 = 4, 𝑟 = 2.
Thus, number of ways = 𝑃 4,2 =

4!
4−2 !
= 12.
In example 1, the Combinations formula applies with 𝑛 = 4, 𝑟 = 2.
Thus, number of ways = 𝐶 4,2 = 2!
4!
4−2 !
= 6.
N.B.

In both cases, a choice of 2 people from 4 people is made.

However, in case 1, the order of choice of 2 people does matter (since the one
person chosen is chairman and the other one vice-chairman).

In case 2, the order of choice of 2 people does matter. In this case, the only
interest is in who serves on the committee.

Hence, there are more permutations than combinations.
Exercise

Find the number of ways to take 4 people and place them in groups of 3
at a time where order does not matter.
Use your calculator

You must study the calculator instructions notes on how to use a calculator
and you can be able to do these calculations using your calculator.