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Counting and Counting Principle

→ 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:

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.

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.

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