## → Definiton:

**Counting **refers to the process of determining the number of elements in a set or the number of possible outcomes in a certain scenario.

• It is used in Inventory Management, Probability, Statistical Analysis etc.

**Combinatory **is a branch of discrete mathematics that focuses on studying arrangements, selections, and combinations of objects from finite sets.

• It is used in Permutations and Arrangements, Poker and Card Games, Coding and Cryptography etc.

## → Basic Counting principles:

## Sum Rule Principle

The **Sum Rule Principle**, also known as the **Addition Principle**, is a fundamental counting principle in combinatorics.

• It states that if there are “**m**” ways to do one task and “**n**” ways to do another task, both of which cannot be done simultaneously, then there are “**m + n**” ways to choose one of these tasks.

• When facing a choice between two mutually exclusive options (A and B), the total number of ways to make the choice is the sum of the number of ways for each option i.e (**m+n).**

**For example:**

• If you can wear either a redshirt(**A**) in 5 ways(**m**) or a blueshirt(**B**) in 3 ways(**n**), then the total number of ways to choose a shirt is**(m+n)**= 5 + 3 = 8 ways.

• If you can travel to a destination either by car in 10 ways or by train in 6 ways, then the total number of ways to travel is 10 + 6 = 16 ways.

## Product Rule Principle

The **Product Rule Principle**, also known as the **Multiplication Principle**, is a fundamental counting principle in combinatorics.

• It states that if there are “**m**” ways to do one task and “**n**” ways to do another independent task, then there are “**m * n**” ways to perform both tasks together.

• If a task can be accomplished by performing operation A in “m” ways and operation B in “n” independent ways, then the total number of ways to perform both tasks is “**m * n**.”

**For example:**

• If you have** 4 shirts** and **3 pairs of pants**, then the total number of outfits you can create by choosing one shirt and one pair of pants is **4 * 3 = 12 outfits.**

• If you can order a pizza with 5 different toppings and a drink with 4 choices, then the total number of meal combinations is **5 * 4 = 20 combinations.**