This is the first of my series of post on combinatorics, a fascinating field that is used to model many real life scenarios. In my opinion, combinatorics attempts to model real life situations more so than many other topics in mathematics.

I will start with the most basic of concepts and move onto more advanced topics as we progress.

A permutation represents the number of different ordering of elements in a row. For example, the set of elements {a,b,c} has the following six permutations:

abc, acb, bac, bca, cab, cba

As you can see, there are six possible ways to order the set {a, b, c} but is there a mathematical formula to model this process? One can imagine that having to count each possible permutation could become a laborious task. Well in order to model this behavior, consider the following example:

Because we have three positions, each position can be thought of as having a certain number of choices. Position 1 can be occupied by three different choices (the full set), while position 2 can be occupied by only two different choices (full set – element at position one), and position 3 only has one choice.

Because of the multiplication rule, we can get the total number of different orderings of elements by just multiplying terms. So the formula for permutation is just n-factorial which makes sense because this expression models the exact process of placing terms until there is only one more term to place.

The above process of finding the number of permutations works if we want to permute an entire set, but what if we want to find the number of 2-permutations in a set of three. This would represent the number of different ways the entire set can be grouped together in sets of two. If we were to find the number of 2-permutations of our original set {a, b, c} it would look as follows:

ab, ba, ac, ca, bc, cb

Once again we come to the number six which is kind of interesting if you think about it. The number of different ways to arrange three terms in a set of three is identical to the number of ways to order two terms. This type of permutation is called an r-permutation and it represents the number of ways a set of r elements can be arranged from a greater set of n elements where Can you take a minute to think about the last restriction we placed on r? Well obviously r cannot be a fraction (no fractions of elements), and it has to be less than n, you can’t have a set of n+1 elements if there are only at most n elements in a set.

The formula for r-permutations is as follows:

This formula can be easily understood if you envision a row if terms getting ready to be multiplied together in order to get the full permutation and then dividing out the number of terms that will leave only the first r terms. In the next post I will give more examples and talk less about theory.