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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.

When we talk about combinatory problems like the socks in the drawer, we're referring to the number of ways certain items can be selected or arranged. This particular problem involves combinations, a fundamental concept in combinatorics, where the order of selection does not matter. In other words, picking a red sock first and a blue sock second is the same as picking a blue sock first and then a red sock.

To understand this on a deeper level, imagine you have a variety of socks in the drawer, and you want to know how many unique pairs you can create. This is where combinatorial formulas, including those for permutations and combinations, come to play. In our example with the socks, since the order of drawing socks does not matter, we use the combination formula to find out the possible pairs.

When solving combinatorial problems, we often use factorial notation, denoted by an exclamation point (!). Factorial notation is a mathematical expression used to describe the product of an integer and all the positive integers below it. For example, saying '5 factorial' is the same as saying the product of 5 times 4 times 3 times 2 times 1, denoted as 5!.

However, by convention, 0! is always equal to 1. This is a fundamental rule in combinatorics that facilitates calculations, especially when working with the combination formulas, where 0! appears frequently.

In reference to our sock example, understanding how to calculate factorial is crucial. You'd calculate the factorial of the total number of socks (n!) and of smaller groups, like the number of red socks (3!), to figure out the different ways to pick the socks. Exploring large numbers quickly reveals how rapidly factorials can grow, which is a fun (and sometimes daunting) feature of this concept.

Probability is all about measuring the chance of an event occurring, usually expressed as a fraction or percentage. Think of it as trying to predict the likelihood of pulling a red sock from a drawer without looking. To calculate basic probability, we need two main components: the number of favorable outcomes and the total number of possible outcomes.

Using our exercise, the event we care about is drawing two red socks. The number of favorable outcomes is the number of ways to choose 2 socks from the 3 red ones available. The total number of outcomes is how many different pairs of socks could be pulled from the drawer. So, the probability is a ratio of these two numbers.

With the concept of basic probability, students can tackle a wide array of real-life scenarios beyond theoretical exercises. From predicting weather events to making informed decisions in games of chance, understanding probability is a valuable life skill that goes hand in hand with combinatory logic and factorial notation.