What are the five stages of counting?

Counting is a crucial part of Fundamental Mathematics. Basic mathematics and all mathematical formulas are derived from numbers and their relations they hold amongst them. Probability is the Mathematical branch dealing with chances and likeliness of a particular outcome out of all the possible outcomes that can occur. A Probability Event gives the measure of chances of Favorable /desired /wanted outcome to be the final outcome of the probability event. Probability largely deals with numbers and the chances of occurrence of different numbers. 

Stages of Counting

There are five basic stages of counting. Starting from Count all, Count on, Maintain cardinality, stable order, and conservative. Let’s take a look at all these stages in detail,

• Count All

Count all the entities irrespective of the order of the entities, keeping in-tact the abstraction of different entities. Entities are considered equal while counting, there is no discrimination done based on the shape, size or characteristics of entity.

• Count On 

Group entities as and when required according to the ask and need of the problem statement. One-to-one Mapping is followed while counting different entities.Each entity that needs to be considered, is only counted once.

• Maintain Cardinality

Cardinality is important w.r.t counting. Recounting should not be practiced, each entity needs to be counted only once. There needs to be no recounting with respect to any of the given entity. 

• Stable Order

By stable order, it is meant that the total number of entities must be a maximum numerical value. Order Irrelevance is the most important stage while considering counting different entities i.e. No fixed order is followed, while counting different entities

• Conservation

The count of entity remains the same, irrespective of the proximity of different entities. The distance between the entities is irrelevant  while  considering  count of different entities. Entities which are placed far will have same count as entities which are placed closer to each other, if they are same in number.

Counting in Probability

Counting in Probability stands for considering the count/ different ways of choices or options available to choose amongst the given entities. The count of choices is determined by the available entities. 

The different stages of Counting deal with the Fundamental Principle of Counting. The Fundamental Principle of Counting thus is very important aspect considering the probability of available choices for numbers. 

The Fundamental Principle of Counting says that for every available entity having ‘n’ choices related with it, then the total number of ways linked to it will be,

n₁ × n₂ × n₃ × n₄…

Steps in Fundamental Principal of Counting

• Step 1: The entity of finding probability is identified.
• Step 2: For each entity the available options are considered.
• Step 3: Each of the available options of the different entities are multiplied, to give the different number of ways of Probability ,or the Count of Number of Different Ways of Probability of a given event.

Sample Problems

Question 1: A teacher has 10 White chalks and 3 Slate Boards. In how many ways can she select a chalk and slate board ?

Solution: 

Choices available for White chalks = 10

Choices available for Slate Boards = 3

Number of Ways for Choosing can take place by the teacher = Choices available for White chalks x Choices available for Slate Boards

= 10 × 3

= 30

Question 2: A boy has choices for 8 Sausages and 4 types of Bread. In how many ways can she select one Sausage an one type of Bread ?

Solution:

Choices available for Sausages = 8

Choices available for Bread = 4

Number of Ways for Choosing can take place by the Boy = Choices available for Sausages x Choices available for Bread

= 8 × 4

= 32

Question 3: Consider a girl having choices to select among 5 Dresses and 4 Snickers.

Solution:

Choices available for Dresses = 5

Choices available for Snickers = 4

Number of Ways for Choosing can take place by the Girl = Choices available for Dresses × Choices available for Snickers

= 5 × 4

= 20

Question 4: A teacher has 27 notebooks and 2 pen: red and blue. In how many ways can she select a notebook and pen ?

Solution:

Choices available for notebooks = 27

Choices available for pen = 2

No. of Ways for Choosing can take place by the teacher = Choices available for notebooks x Choices available for pen

= 27 × 2

= 54

Question 5: Consider a couple having choices to select among 8 Starters and 6 Desserts for Dinner.

Solution:

Choices available for Starters = 8

Choices available for Desserts = 6

Number of Ways for Choosing can take place by the couple = Choices available for Starters × Choices available for Desserts

= 8 × 6

= 48

Question 6: Consider a girl having choices to select among 10 Dresses and 15 Hairbands.

Solution:

Choices available for Dresses = 10

Choices available for Hairbands = 15

Number of Ways for Choosing can take place by the Girl = Choices available for Dresses × Choices available for Hairbands

= 10 × 15

= 150

Question 7: Consider a girl having choices to select among 3 Dresses and 2 Snickers.

Solution:

Choices available for Dresses = 3

Choices available for Snickers = 2

Number of Ways for Choosing can take place by the Girl = Choices available for Dresses × Choices available for Snickers 

= 3 × 2

= 6