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Combinatorics is the branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, to name a few. One key aspect of combinatorics is the concept of counting and arranging elements within a set according to certain rules, which leads us to concepts like permutations and combinations, which are fundamental in calculating probabilities in games like poker.

In the context of a poker game, combinatorics can help players understand the different possible hands they can be dealt. For example, when considering a straight flush, combinatorics allows us to calculate how many different ways we can select 5 consecutive cards of the same suit from a standard 52-card deck.

Factorial notation is incredibly useful when it comes to calculations involving combinatorics. A factorial represents the product of all positive integers up to a certain number, denoted by the number followed by an exclamation point. For instance, the factorial of 5, written as '5!', is calculated as 5 x 4 x 3 x 2 x 1, which equals 120.

This notation is especially important in the framework of permutations and combinations, where we often need to calculate factorials of large numbers to determine the total number of possible arrangements or selections. In the scenario of our poker hand, the factorial notation helps us to efficiently calculate the number of different ways to draw 5 cards from 52.

Permutations and combinations are two fundamental principles of combinatorics that deal with the arrangement of items. Permutations refer to the arrangement of items where the order is important, while combinations refer to the arrangement where the order does not matter. In poker, the order in which the cards are dealt is not important; thus, we use the concept of combinations.

To illustrate, the number of ways to get any 5-card hand from a 52-card deck is given by a combination, since the order of cards in your hand does not change the value of the hand. The formula for a combination is given by 'n choose k', which calculates the number of ways to choose k items from n options without regard to the arrangement's order.

Probability theory is the part of mathematics that studies random phenomena. It provides a way to quantify the uncertainty of events, using values between 0 and 1, with 0 representing the impossibility of an event and 1 representing certainty. Applying probability theory to card games like poker allows players to compute the likelihood of being dealt certain hands.

In the context of our exercise, the probability of being dealt a straight flush can be determined by dividing the number of straight flush hands by the total number of possible poker hands. This calculation gives us a numerical measure of how likely it is for a player to receive a straight flush in a single deal of a five-card poker hand.