Permutations and Combinations
The Permutations and Combinations category is often called Combinatorics, which is the branch of mathematics dealing with permutations & combinations. It begins with the Fundamental Counting Principle; the basic principle of this unit.
Permutations (arrangements where order is important) are covered in great detail. There are linear permutations (arrangements in a straight line), permutations with no restrictions, with restricted objects, with adjacent objects, and with repetitions. Circular and Ring permutations are also covered in this category.
The next topic is combinations (selections where order is not important). Questions are done using coins, committees, playing cards, polygons, etc.
Pascal's Triangle is then introduced which leads into Pathways.
The category ends with the Binomial Theorem, expanding (a+b)n. Here the connection is made between expanding a binomial & using combinations.
The Fundamental Counting Principle
This lesson is the beginning of the Permutations & Combinations lessons. Using the fundamental counting principle is the basis for understanding how we deal with permutations & combinations.
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Factorial Notation & Linear Permutations
The first part of this lesson introduces & works with factorial notation by simplifying expressions & eventually solving equations with factorials.
The second part of this lesson deals with the definition of a permutation & the formula for a permutation.
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Linear Permutations with Restrictions
This entire lesson works with permutations which have restrictions on the objects in the question. It proceeds from basic restrictions to adjacent objects and then to repetitions.
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Circular & Ring Permutations
This lesson introduces the concept of a circular permutation & how it differs from a linear permutation.
The ring permutation, which is a circular permutation that can be flipped or reflected, is also covered in this lesson.
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Combinations Part I
This lesson is all about combinations and the difference between a combination and a permutation. Questions are done using the combination formula involving numbers in lotteries, people on committees, cards in a standard deck, sums of money, etc.
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Combinations Part II
: This lesson is a continuation of Combinations Part I. Questions are done with non-collinear points, polygons & diagonals, more difficult problems with cards and numbers on committees.
Also covered in this lesson are selections from the same group, divi
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Pascal's Triangle and Pathways
This lesson is an investigation of Pascal’s Triangle & how it is used to find the coefficients in the expansion of a binomial. Coefficients are also found using combinations. The relationship is also shown between Pascal’s Triangle and Pathways.
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The Binomial Theorem
This lesson is the study of the Binomial Theorem. It begins with the expansion of a binomial and then moves on to finding a specific term, finding the middle term, and ends with finding the constant term.
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