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Principle of Counting

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While dealing with the recurrence function, we have to solve the function with different methods and approaches and we also apply some principles to solve them. Here, we have the basic principle of counting. That is, if we have p ways to do one thing and q ways to do other thing, then there are total p x q ways of doing both. This is the basic counting principle. It is a way to figure out the total number of ways, different events can occur and is all about choices we might make given many possibilities.

A fundamental counting principle, can be a method helpful to calculate each of the possible combination of particular quantity of events.
For example, consider that we need to buy a new car and there are two different colors available with different features and functions. Then, we have several choices to make our life comfortable. This is the basic counting principle that we use in our daily life to make choices in different situations and conditions, to make smart choices. Here, one is free to choose the best condition and that defines the fundamental principle of counting. The process of counting is necessary and ideal in the solution of recurrence functions as these counting can predict the solution approximately or the exact one.

Addition Principle of Counting

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Let $E_{1}$ and $E_{2}$ be disjoint events with $O_{1}$ and $O_{2}$ possible outputs respectively. Disjoint events are the events that have no common outputs. Then, total number of outputs for event "$E_{1}$ or $E_{2}$" is $O_{1}$ + $O_{2}$.
It is to be noted here that events must always be disjoint. In other words they must not have common outputs. Otherwise addition principle of counting is no longer applicable.

This addition principle of counting can be applied on more than two events.
When more than two events are used then this principle can be explained as follows
Let $E_{1}$ , $E_{2}$ ,... , $E_{k}$ be events with $O_{1}$ , $O_{2}$ , ... $O_{k}$ possible outputs respectively. Then total number of outputs for event "$E_{1}$ or $E_{2}$ or ... or $E_{k}$ " is $O_{1}$ + $O_{2}$ + ... + $O_{k}$. Here it is to be noted that events are disjoint.

We can understand concept of addition principle of counting with help of an example as follows:
Suppose we are going to purchase a mobile handset from one of the two companies X and Y. Assume that these companies manufacture 15 and 14 different models of mobile handset respectively. Then, total number of models altogether to choose from can be counted using addition principle of counting.
It is explained as follows:
Since we can choose one of 15 models of company 'X' or one of 14 models of company 'Y', hence there are altogether 15 + 14 = 29 models among which one should be selected. This is called as Addition Principle of Counting.

Multiplication Principle of Counting

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When we deal with happening of more than one event or activity, then it is necessary to quickly confirm that how many possible outputs exist. Number of possible combinations of all events to occur can be obtained by counting principle.

In a simple form, counting principle can be understood in the following way:
If one activity has ‘n’ ways to occur and another activity has ‘m’ ways to occur, then there are ‘n x m’ ways for both to occur.

Let us understand it by rolling a die and flipping a coin. If there are 4 ways to roll a die and 3 ways to flip a coin.
Then, there will be 4 x 3 = 12 possible ways to roll a die and flip a coin.

Counting Principle can also be applied on more than two activities or events.
For example, if a coin is tossed five times and there are 3 ways to flip each coin, then 3 x 3 x 3 x 3 x 3 = 243 are the possible arrangements of heads and tails.

Let us try to understand properly the multiplication principle of counting using the following example:
For our breakfast, if we make a cheese sandwich or a pizza slice, then we have two choices. And, if we have either apples or cookies for dessert, then also there are two choices. According to multiplication principle of counting, there will be a total of 2 x 2 ways to have our dinner and dessert.

The 4 different possible ways are as shown below:
  1. Cheese sandwich & apples
  2. Cheese sandwich & cookies
  3. Pizza slice & apples
  4. Pizza slice & cookies