## Combinations and Permutations

### Jul 23, 2014

Many people find this topic difficult. I am interested in understanding why it is difficult and how to address this challenge. This will be a series of posts covering the very basics of this topic leading to links to much more thorough and detailed treatments.

An obvious place to start is to try to determine what combinations and permutations are. In good modern mathematical methodology I suppose I describe what they **do** rather than what they **are**.

So I understand combinations and permutations to be the techniques applied to situations of counting, selecting and arranging.

Ok, time for definitions of: counting, selecting and arranging.

Counting. Well we all know what that is, don’t we? Ok, define counting. I will find some links and maybe alter my definition but off the top of my head I understand it to be the process of associating (linking? mapping?) elements between two sets where element a_{i} in A is uniquely mapped to element b_{i} in B. The two sets are said to have the same number of elements or size (cardinality?). For example, {A, B, C} has A > 1, B > 2, C > 3, 3 elements.

That isn’t hard. Well, quite quickly it can be when we add situations that require selecting objects. What is hard? Well, quickly the numbers can be come big even though the actual numbers we are arranging is quite small (see soon for an example). And our original counting approach of listing out all possible objects becomes tedious because of the number of objects involved or we lose confidence that we have counted all the objects or maybe we have counted some objects twice. but, whoa. How can all this difficulty arise? Counting is easy: A > 1, B > 2, C > 3 .

Maybe a hint about counting infinities here or numbers to measure relative size rather than count. Maybe how to extend counting to basic arithmetic operations.

Time for some examples:

An explosion numbers < 4 distinct objects> combinations and perms

Increasing complexity < repeated, identical, non-distinct objects >

to continue …..