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The factorial operation is a mathematical function symbolized by an exclamation mark, such as \( n! \). It represents the product of all positive integers less than or equal to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials are especially useful in permutations, combinations, and other areas of mathematics. To give another example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

For any number \( n \), the factorial grows very quickly. Remembering the definition is key for working with more complex expressions involving factorials.

Simplifying expressions often involves reducing them to their simplest form. In the given exercise, the goal was to evaluate \( \frac{7!}{(7-2)!} \). By simplifying the denominator first, we have \( (7-2)! = 5! \), which makes our expression \( \frac{7!}{5!} \).
From there, we can expand each factorial: \( 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) and \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \). By canceling out the common terms in the numerator and the denominator, we are left with \( \frac{7 \times 6 \times 5!}{5!} = 7 \times 6 \). So, the simplified result is 42. Reducing expressions in this way makes complex problems easier to solve.

Combinatorics is a branch of mathematics dealing with combinations, permutations, and counting. Factorials play a crucial role in these calculations. A common example is finding the number of ways to arrange objects, called permutations. The number of permutations of \( n \) objects is calculated as \( n! \).
For example, the number of ways to arrange 4 books on a shelf is \( 4! = 24 \). Another concept in combinatorics is the combination, which determines how many ways you can choose \( k \) objects from a set of \( n \) objects, usually denoted as \( C(n, k) \) or \( \binom{n}{k} \).
This is calculated as \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]. For instance, choosing 2 fruits from an assortment of 7 is calculated as:
\[\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7!}{2!5!} \inserting \; factorials,\; we\; get \frac{7\times6}{2\times1} \].
This simplifies to 21 ways. Combinatorics provides tools for solving real-world problems involving arrangements and selections.