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Evaluating expressions in mathematics is about replacing variables with their numerical values and simplifying the result following the order of operations. Let's take the given expression \(\frac{{ }_{20} P_{2}}{2 !}-{ }_{20} C_{2}\). To evaluate it, we need to compute each part separately using the formulas for permutations and combinations.

Firstly, let's deal with the permutation \( _{20}P_{2} \) which involves arranging 2 items out of 20. The permutation formula \( P(n, r) = \frac{n!}{(n-r)!} \) gives us a clear path to follow. By plugging in the numbers we get \( _{20}P_{2} = \frac{20!}{18!} \). Then, we tackle the combination term \( _{20}C_{2} \) which involves selecting 2 items out of 20 without considering order. Using the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \) and substituting the values, we find that \( _{20}C_{2} = \frac{20!}{2! \cdot 18!} \).

The final step is to execute the operations as expressed. You subtract the evaluated combination from the evaluated permutation divided by 2 factorial, simplifying to get your final answer. Remember, there's no magic to it—just follow the formulas and mathematical order of operations step by step.

Factorial notation is a shorthand way to represent the multiplication of a sequence of descending natural numbers and is denoted with an exclamation point (e.g., \(n!\)). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1\). It's crucial in permutations and combinations as it helps quickly calculate the number of ways in which a set of items can be arranged or selected.

In our example, \(20!\) would be the product of all natural numbers from 20 down to 1. Factorial notation is particularly useful as it simplifies to \(\frac{20!}{18!} = 20 \times 19\), since the \(18!\) terms cancel out. Similarly, for combinations, it helps to simplify expressions like \(\frac{20!}{2! \cdot 18!}\) which reduces to \(\frac{20 \times 19}{2 \times 1}\), by canceling out the \(18!\) terms. It's a powerful tool that makes calculating large expressions more manageable and less time-consuming.

Mathematical reasoning involves the process of using logic to figure out the truth of a statement based on known facts and relationships. In the context of permutations and combinations, it's the understanding of why formulas work and how they apply to a given problem.

When we examine the expression in the provided exercise, mathematical reasoning guides us to understand that a permutation is concerned with the order of selection, which is why it has more possibilities than a combination for the same number of items. This reasoning is supported by the formulas where the permutation term doesn't have an \(r!\) in the denominator, indicating that order matters.

By applying mathematical reasoning to evaluate the given expression, we can assert that the difference between the permutation and combination values will give us the extent by which the ordering of elements affects the total arrangements. Thus, continued practice in reasoning will enhance your ability to dissect complex problems and choose the right mathematical tools to solve them efficiently.