Understanding Permutations and Combinations for AI

Once upon a time, in a quiet town there was a bakery owned by a kind hearted baker, named Mr. White. He had three types of delicious pastries, chocolate, strawberry and vanilla. As part of his sales, he had a special box of pastries where a customer could choose how many pastries to include and which pastries.

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There were a few conditions:

1. The customer could choose two to four pastries.

2. He could select any combination of flavors including duplicates.

3. The order of the pastries did not matter.

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How many different combinations could the customer choose? Suppose the customer chooses two pastries. Since there were three types and repetition was allowed, the customer had three choices for the first and three choices for the second.

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Hence, the customer had 9 possibilities. Likewise, if the customer chose three pastries, then there were twenty seven possibilities . For a box of four pastries, a customer would have 81 possibilities.

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So the customer had 9+27+81=117 possibilities since he could choose two or three or four pastries. Note that when it is "OR' we add. This is how, Permutations and Combinations can make a seemingly complex mathematical concept come to life.

Welcome to this course on Permutations and Combinations. I am Suman Mathews, qualified and experienced teacher teaching mathematics for three decades to high school and college students.

Permutations and Combinations forms an integral part of higher Mathematics. It is used extensively in Probability, Binomial Theorem and Discrete Math.  Also used in SAT and GRE Quant examinations. The tricky part here is that each problem is unique.

There is no one solution to all problems on Permutations and Combinations. This course will introduce you to the basics and gradually lead you forward. Make use of the solved examples and resources here.

The course unfolds with a basic knowledge of Permutations. You will learn cases when the objects are all different or not all different.  Learn basic formulas in Permutations here.

Number of problems involving letters and digits are discussed here . Moving on, you'll learn about circular permutations and how to calculate them. You'll realise how the values change in a circular permutation where no two people have the same neighbours or different neighbours.

Again, practice the problems illustrated here to gain mastery of the topic. Next, you'll learn about Combinations and how to evaluate them. Realise that while a permutation is basically an arrangement, a combination is a selection.

Learn basic properties of Combinations and how to use these in problem solving. You'll come across problems which use both Permutations and Combinations and work on mixed problems. You'll realise how these are interconnected.

You have a session on Multiple choice questions where you realise that each problem requires a fair level of understanding. There are also formulas at the end for you.

Learn higher order problem solving techniques in Permutations.