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Factorials are a fundamental concept in combinatorics and are denoted by an exclamation mark (!). For example, the factorial of 6 is written as \(6!\). The factorial of any positive integer \(n\) is the product of all positive integers less than or equal to \(n\). This means that \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).

Factorials are important when calculating permutations and combinations because they help in determining the total number of ways to arrange or choose items. One special case is \(0!\), which is defined to be 1. This might seem counterintuitive at first, but it ensures consistency in various mathematical formulas, including those for combinations and permutations.

The combination formula allows us to calculate the number of ways to choose \(r\) items out of \(n\) available items, without considering the order in which they are selected. The formula is given by:

\[C_{n}^{r} = \frac{n!}{(n-r)!r!}\]

The formula involves factorials, which appear in the numerator and the denominator. Here's a quick breakdown:

• \(n!\) represents the factorial of the total number of items.
• \((n-r)!\) represents the factorial of the difference between the total items and the selected items.
• \(r!\) represents the factorial of the number of items being chosen.

The beauty of the combination formula is that it calculates selections where the order does not matter. If you were arranging them, you would use permutations instead. In the given exercise, when calculating \(C_{6}^{6}\), after plugging in the values, it simplifies because \((6-6)!\) is \(0!\), which equals 1.

The binomial coefficient is another term for combinations and is written as \(C_{n}^{r}\) or sometimes \(\binom{n}{r}\). The binomial coefficient represents the number of ways to choose \(r\) items from \(n\) items without regard to the order of selection.

The binomial coefficient is a critical component of the binomial theorem, which is used to expand expressions of the form \((a + b)^n\). The terms in a binomial expansion are in the form of \(C_{n}^{r}\), showing how many ways each part of the expansion occurs.

In our exercise, \(C_{6}^{6}\) represents the scenario where we choose all 6 items from a set of 6, resulting in only one way to do so (hence, the result is 1). This exercise illustrates the simplicity of the binomial coefficient when \(r\) equals \(n\) because all elements are selected, thereby reducing complexities in calculations.