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Understanding factorials is essential when working with combinatorics. A factorial of a non-negative integer, say 'n', denoted by 'n!', is the product of all positive integers less than or equal to 'n'. For instance, the factorial of 5 is calculcated as follows:
\[\begin{equation}5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.\end{equation}\]
This concept becomes particularly useful when dealing with the number of ways things can be arranged or combined. Interestingly, the factorial of 0 is defined to be 1, which is convenient when formulae like the binomial coefficient are involved. A common mistake is the misinterpretation of factorials - it's crucial to remember that factorials grow extremely rapidly and are only defined for non-negative integers.

Combinatorics is a field 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, to name a few. The binomial coefficient is a pivotal concept in combinatorics and represents the number of ways to choose 'r' elements from a set of 'n' elements without regard to the order of the elements. A fundamental aspect to grasp is the difference between combinations and permutations, as the latter takes into account the order of the elements.

When we discuss permutations and combinations, we're delving into the different ways of arranging and selecting items. Permutations are concerned with the arrangement of objects: 'How many different ways can you arrange a set of items?'. In permutations, the order does matter. Conversely, combinations refer to the selection of items where the order does not matter. This relates directly to the binomial coefficient formula:
\[\begin{equation}_{n}C_{r} = \frac{n!}{r! (n - r)!}.\end{equation}\]
For our example, \[\begin{equation}_{18}C_{2}\end{equation}\] is calculated by dividing the factorial of 18 by the product of the factorial of 2 and the factorial of 16 (which is 18 minus 2). It tells us how many ways we can select 2 elements from a set of 18. Remember, in combinations, because the order of selection doesn't matter, '_{18}C_{2}' and '_{18}C_{16}' will yield the same result due to the symmetric property of binomial coefficients.