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Counting Principle

In mathematics, we sometimes come across with a concept known as the "counting principle". This is concerned with the total number of combinations.

When we deal with the happening of two or more activities or events, it is often required to know quickly the number of total possible activities or events. It can be done by counting principle.

This principle belongs to the topic "permutations and combinations" from algebra - a branch of mathematics. Counting principle is quite commonly used in problems related to combination.
The basic idea of counting principle is that if there are particular number of ways of performing one action and some number of ways of performing another action, then total number of performing both actions would be equal to the product of both types of ways. The counting principle is also known as fundamental principle of counting or multiplication principle or rule of product. This principle is also used in probability theory and in number theory.In the page below, we shall discuss about the counting principle and its applications. So, go ahead with us and learn about this concept in detail.

 

Basic Counting Principle

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The general counting principle is known as the basic counting principle or the fundamental counting principle. With the use of this principle, one can easily determine the total number of ways a combination of two events can be performed.

According to basic counting principle:If an event can be done in some known number of ways and another event can also be completed in some other number of ways, then the total number of ways both events are performed is equal to the product of both the counts, i.e. equal to the product of the number of ways both events can be done separately.  

In other words, we can say that:
If there are lets say "p" number of possible ways for one type of action; also there are "q" ways for another type of action, then the number of ways of performing both of them would be equal to the product of p and q; i.e. p x q.

For Example: If a girl has 5 skirts and 7 tops, then she can actually try 5 x 7 ,i.e. total 35 number of combinations.

Counting Principle in Probability

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In probability, the counting principle is utilized in order to find the total number of distinct ways in which a certain event can be performed.

The counting principle can be understood in the form of following two rules in the field of probability theory.

1) Assume that there are n numbers of events that are distinct from one another. Lets say these events are denoted by A$_{1}$, A$_{2}$, A$_{3}$, ...., A$_{n}$, then the total number of sequences which can happen from these given n events would be :
A$_{1}$ . A$_{2}$ . A$_{3}$ ... A$_{n}$

For Example: We have a set of letters P, Q, R and another set of letters A, B; then there would be 3 x 2 = 6 possible combinations available.
i.e.
PA, PB, QA, QB, RA, RB

2) Let us suppose that there are k number of mutually-exclusive (the events that cannot occur simultaneously) and exhaustive events (the events that have fixed known outcomes). These events are performed a certain number of times; such as n trials of such events. In this case, there will be $k^{n}$ number of sequences which can be obtained with such trials.

For Example:  If a coin is flipped four times, then total number of possible events would be equal to $2^{4}$ i.e. 16. Here, k = 2 since there are 2 exhaustive and mutually-exclusive events - head and tail. Also, n = 4 since there are 4 flips of the coin.

For instance - if we want to order a sandwich. We may select bread among brown bread, multi-grain bread and a normal bread. Also, two types of filling veg or non veg are available. In this way, we will have 3 x 2 = 6 types of combinations.

Therefore, the Counting Principle is used to calculate the number of different and unique combinations of performing some task. So, this principle is an essential concept of probability and a fundamental mathematical idea.

Counting Principle in Set Theory

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The counting principle is also used in set theory. In set theory, this principle is often defined as the product of cardinal numbers (natural number including zero; i.e. 0, 1, 2, 3, 4, ...). If {C$_{1}$, C$_{2}$, ..., C$_{n}$} be the a set of n cardinal numbers (This set is not necessary to be a finite set), then we may write that
|C$_{1}$| . |C$_{2}$| .... |C$_{n}$| = |C$_{1}$ x C$_{2}$ x ....x C$_{n}$|
Here, the operator "x" is said to be the Cartesian product operator. The Cartesian product operator is defined as a mathematical operator that gives a product set out of given multiple sets. In other words, if two sets P and Q are given, then Cartesian P x Q will be is a set of all possible ordered pairs, i.e. (p, q), in which p $\in$ P and q $\in$ Q.

Problems

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Few sample problems based on counting principle are given below.
Problem 1: A school management has chosen 5 boys and 6 girls to represent the annual function of the school. Among them, a pair of boy and girl would be randomly selected for final anchoring of the function. Determine the total number of pairs in order to choose one.

Solution: According to counting principle, the total number of pairs will be equal to the product of the total number of boys and girls.
Number of possible pairs = Number of boys x number of girls
Number of possible pairs = 5 x 6 = 30
There would be 30 pairs among which 1 has to be selected.

Problem 2: Find the total number of possible outcomes is a die has to be rolled four times.

Solution : The number of possible events is 6 because a die can be rolled in 6 ways. i.e. the possible event is each roll are getting 1, 2, 3, 4, 5, 6.
According to the counting principle
Total number of possible outcomes = 6$^{4}$
Total number of possible outcomes = 6$^{4}$ = 1296

Problem 3: On a dinner party, James found 2 types of soups, 4 types of burgers, 3 types of chocolates, 5 types of beverages and 4 types of desserts. Calculate how many combinations of foods James can have ?
Solution: Number of soups = 2
Number of burgers = 4
Number of chocolates = 3
Number of beverages = 5
Number of desserts = 4

Applying counting principle,
Total number of foods = 2 x 4 x 3 x 5 x 4
Total number of foods = 480.
More topics in Counting Principle
Fundamental Counting Principle
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