Combinatorics

COMBINATORICS

Variations permutations and combinations

Strategies based on the product

Historic references

Combinatoric numbers

Historic references

Blaise Pascal Pascal's contributions to probability theory, developed through his correspondence with Pierre de Fermat, include foundational concepts that are central to the field. The key idea they developed is the concept of expected value and the principles underlying it.

The Tartaglia's triangle Tartaglia's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. It is named after the Italian mathematician Niccolò Fontana Tartaglia, who studied it in the 1 6th century. In this triangle: - The topmost element (the apex) is 1. - Each element is obtained by adding the two elements above it. - The edges of the triangle are always 1. - This triangle has various applications in combinatorics

Strategies based on the product

The tree diagram strategy in math visually represents all possible outcomes of a sequence of events by branching out choices at each step, helping to organize and count possibilities.

The Pigeonhole Principle is a simple yet powerful concept in combinatorics. It states that if you have more items than containers, at least one container must hold more than one item.

Variations, permutations and combinations

Variations with repetition

Variations without repetition

Use variations with repetition when you need to arrange a subset of items from a larger set, where the order matters, and items can be repeated.

Use variations without repetition when you need to arrange a subset of items from a larger set, where the order matters, and each item can only be used once.

Permutations

Combinations

Use combinations when selecting items from a set where order does not matter.

Use permutations when you need to arrange all items from a set in a specific order and to use all the elements

Combinatoric numbers

A combinatoric number is used in mathematics to count the different ways to select k elements from a set of n elements without considering the order of selection. This is fundamental in combinatorics and is denoted as , which is read as "n choose k" or "combinations of n elements taken k at a time." The combinatoric number is an essential tool in mathematics for counting combinations without considering order. The formula allows these values to be calculated efficiently, facilitating their application in a variety of mathematical and practical contexts.

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