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Understanding factorials is essential when working with combinatorics and especially when calculating binomial coefficients. Simply put, a factorial of a non-negative integer, say n, is the product of all positive integers less than or equal to n. This is represented as n!. For example, the factorial of 5, denoted as 5!, is calculated by multiplying all the whole numbers from 5 down to 1:
5! = 5 × 4 × 3 × 2 × 1 = 120.
The operation may seem straightforward, but factorials grow exponentially, meaning the numbers can get very large quite quickly. Interestingly, the factorial of zero, 0!, is defined to be 1. This may initially appear counterintuitive, but it plays a crucial role in combinatorics and ensures that the formulas used in permutations and combinations hold true for all applicable values.

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 related to many other areas of mathematics, such as algebra, probability, and geometry, and is widely used in fields such as statistics, computer science, and physics.
The subject deals with the various ways of arranging and grouping objects without considering the order (combinations) or with considering the order (permutations). In our binomial coefficient example, combinatorics is used to determine how many ways we can choose 6 items from a set of 10, without concerning their order. This calculation of choices or possible combinations is a fundamental part of the subject and is applicable in real-world scenarios such as predicting probabilities or optimizing systems.

Permutations and combinations are two core ideas in combinatorics that deal with the arrangement of sets of items. Essentially, permutations concern themselves with the arrangement of objects where the order does matter. In contrast, combinations deal with the arrangement of objects where the order does not matter.

Permutations

If you had a set of unique items and wanted to find out how many different ways you could arrange them, you'd use permutations. For instance, consider the arrangement of letters A, B, and C. These can be arranged in 6 different ways (ABC, ACB, BAC, BCA, CAB, CBA). The number of permutations of n unique items taken r at a time is calculated using the formula: nPr = n! / (n - r)!

Combinations

For combinations, the order of those items is irrelevant. To determine how many ways you can choose ‘r’ items from a set of ‘n’ items, you would use binomial coefficients, which are a central part of combinations. The formula for combinations is given as: nCr = n! / [r!(n-r)!], which is exactly what is used to solve the binomial coefficient in our original exercise.