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A factorial is a mathematical concept that involves multiplying a series of descending natural numbers. It is denoted by an exclamation mark (!). For example, 5 factorial (written as 5!) is calculated as 5 x 4 x 3 x 2 x 1 = 120. Factorials are essential for many areas in mathematics, particularly in combinations and permutations. When solving combinations, you often simplify expressions involving factorials, as seen in our exercise. For instance, 12! (12 factorial) is calculated as 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. Understanding the properties and simplifications of factorials makes working with combinations more manageable.

The combination formula is used to determine the number of ways to choose a subset of items from a larger set, without considering the order of selection. The formula is given by: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, \(n\) represents the total number of items, and \(k\) represents the number of items to choose. In Shawn's case, he needs to choose 3 schools out of 12, which we solve using the combination formula: \(\binom{12}{3} = \frac{12!}{3! \times 9!}\). By understanding this formula, you can solve similar problems where the order of choice does not matter. It's crucial to know that combinations differ from permutations, where the order of selection plays a role.

When registering for the SAT, students can choose several colleges to receive their scores at no cost. This selection process can be thought of in terms of combinations. Shawn, our example student, can choose any 3 out of 12 colleges for his scores to be sent to. Using the combination formula, we find that there are 220 different ways for Shawn to pick the colleges. This kind of application highlights the usefulness of combinations in making real-life decisions, helping students understand the practicality of mathematical concepts. Knowing how to calculate combinations can be helpful in many scenarios, from school applications to decision-making in professional and personal settings.