Basic Principle of Counting

BASIC PRINCIPLE OF COUNTING

Chapter 6 of the sources is titled Basic Principle of Counting. It introduces the fundamental mathematical rules used to determine the number of ways events can occur , which serves as a prerequisite for understanding probability.

1. Addition Rule of Counting

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Addition Rule ( $n_1 + n_2$ )

This rule is used when you must choose between different actions that cannot happen at the same time.

If an action $A$ can occur in $n_1$ ways and action $B$ can occur in $n_2$ ways, the total number of ways of occurrence for either $A$ or $B$ is $n_1 + n_2$ .

Example

You have a gift card to buy one item: either a shirt or a pair of trousers.

Shirt choices: 4
Trouser choices: 3
Total choices: $4 + 3 = \mathbf{7}$ ways.

2. Multiplication Rule of Counting

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Multiplication Rule ( $n_1 \times n_2$ )

This rule is used when multiple actions occur together.

If action $A$ can occur in $n_1$ ways and action $B$ can occur in $n_2$ ways, the total number of ways for both $A$ and $B$ to occur together is $n_1 \times n_2$ .

Example

A gift card allows you to buy one shirt and one pair of trousers.

Shirt choices: 4
Trouser choices: 3
Total choices: $4 \times 3 = \mathbf{12}$ ways.

3. Generalisation of the Multiplication Rule

If $r$ actions are performed in a specific order, and there are $n_1$ possibilities for the first, $n_2$ for the second, and so on…

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General Formula

Total possibilities = $n_1 \times n_2 \times \dots \times n_r$

Example: Buying one shirt (4 choices), one pair of trousers (3 choices), and one pair of shoes (2 choices).

Solution: $4 \times 3 \times 2 = \mathbf{24}$ ways.

Solved Practice Examples

QQ1

Password Creation (Repetition Allowed)

Create a six-digit alphanumeric password: the first two characters must be upper-case alphabets and the remaining four must be numbers. Repetition is allowed.

View Detailed Solution ▼

There are 26 alphabets and 10 digits (0–9).

Ways: $26 \times 26 \times 10 \times 10 \times 10 \times 10 = \mathbf{6,760,000}$ .

QQ2

Password Creation (No Repetition)

Same as above, but characters and numbers cannot be repeated.

View Detailed Solution ▼

The pool of available choices decreases after each selection.

Ways: $26 \times 25 \times 10 \times 9 \times 8 \times 7 = \mathbf{3,276,000}$ .

QQ3

Race Finishers

Eight athletes compete in a 100m race. In how many ways can they finish (no ties)?

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Any of the 8 can come first, then 7 for second, 6 for third, and so on.

Ways: $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = \mathbf{40,320}$ .

Unsolved Problems for Practice

The Sports Shop (Either/Or)

Narendra is buying sports items. He has a choice of 4 cricket bats, 4 cricket balls, 8 brands of stumps, 5 brands of jerseys, and 10 brands of shoes. If he can only purchase one item from any of these categories, how many ways can he choose?

View Detailed Solution ▼

This uses the Addition Rule.

$4 + 4 + 8 + 5 + 10 = \mathbf{31}$ ways.

The Sports Shop (One of Each)

Using the same choices as above, how many ways can Narendra purchase one item from each category?

View Detailed Solution ▼

This uses the Multiplication Rule.

$4 \times 4 \times 8 \times 5 \times 10 = \mathbf{6400}$ ways.

Class Elections

A class has 60 students. A cricket captain and a class representative must be elected. If a student can only hold one position at a time, what are the total ways?

View Detailed Solution ▼

Multiplication rule without repetition. 60 choices for the first role, 59 for the second.

$60 \times 59 = \mathbf{3540}$ ways.

All Chapters in this Book

Lesson 1

Statistics

Introduces the subject as the 'art of learning from data,' covering its collection, description, and analysis.

Lesson 2

Data

Focuses on the nature of information itself and how it is categorised.

Lesson 3

Describing Categorical Data

Visualising and identifying the 'centre' of qualitative data.

Lesson 4

Describing Numerical Data

Tools for organising and measuring the typical values and spread of quantitative variables.

Lesson 5

Association Between Two Variables

Explores how information about one variable can provide insight into another.

Lesson 6