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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 and has many applications ranging from logic and algorithmic graph theory to physics, biology, and statistics.

For example, when deciding how many different combinations of items can be selected from a larger pool, combinatorics comes into play. This is often used to analyze possible outcomes in games, to optimize computer networks, or even decide the best layout for planting in a given space. In our exercise, combinatorics helps us determine the number of ways to select 3 items from a set of 54, using the well-known combinations formula. In this context, the exercise teaches students to apply combinatorial reasoning to calculate a specific type of problem known as combinations.

The factorial of a non-negative integer, denoted by 'n!', is the product of all positive integers less than or equal to 'n'. It's a fundamental concept in the field of combinatorics.

In order to simplify the calculation process, most scientific and graphing calculators include a factorial key, usually represented by '!' or 'n!'. To evaluate factorials with a calculator, one typically enters the integer, followed by pressing the factorial key. This functionality allows for efficient computation of large numbers that would otherwise be fairly time-consuming to calculate manually.

For instance, finding the value of 54! would be quite laborious by hand, but with the factorial key, it's a simple matter of entering '54' followed by the factorial key, which aids immensely in our textbook exercise.

The concepts of permutations and combinations are used to solve problems related to counting and are essential parts of combinatorics.

Permutations

Permutations are about finding the number of possible arrangements of a set, where the order does matter. For example, the number of ways to arrange letters or numbers.

Combinations

Combinations, on the other hand, refer to the selection of items from a larger set where the order does not matter. The exercise mentioned considers combinations, exemplified by using the formula to find how many ways we can select 3 items from a set of 54 without regard to order.

Understanding the differences and applications of permutations and combinations is critical for solving various problems in probability, statistics, and more. In summary, permutations focus on sequences, whereas combinations emphasize groups or selections.