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Factorials are a fundamental mathematical concept used in permutations to count the number of ways to arrange a set of items. The notation for a factorial is "!". For example, the factorial of a number, say 5, is written as 5!. This means you multiply all whole numbers from the number down to 1:

• 5! = 5 × 4 × 3 × 2 × 1

This product results in 120. Factorials grow very quickly, as each value is multiplied by one higher number. For instance, 6! is 720 as you multiply 6 by all the numbers before it.
Factorials are particularly useful in calculating permutations and combinations because they help establish the total number of sequences possible, even when the sequences are quite large. Remember, 0! is defined as 1 since there's only one way to arrange zero objects.

Permutations are a specific way to arrange items where the order of arrangement is significant. The formula for permutations is given by:\[P_r^n = \frac{n!}{(n-r)!}\]Here, \(n\) is the total number of items you can choose from, and \(r\) is the number of items to arrange. The expression \(n!/(n-r)!\) represents all the unique sequences of \(r\) items from \(n\) possibilities.
In our example, \(P_1^{20}\), we calculate the permutation of 1 item selected from a set of 20.

• First, multiply all integers from 20 down to 1 to get 20!, then divide this by 19! (which consists of all integers from 19 down to 1).

This division effectively cancels out all terms except for 20, leaving us with the answer of 20. It shows that there's really only one possible outcome when choosing one distinct item from a group, as shown by the remaining factorial calculation.

Mathematical evaluation involves carrying out all the necessary calculations to make sense of a formula and obtain a specific result. In permutations, evaluating the formula involves simplifying the expression through proper use of factorials and cancellation.
For example, simplifying \(\frac{20!}{19!}\), allows us to cancel all common terms in the numerator and the denominator except for what is left as the simplest form:

• 20 alone, which simplifies from the factorial sequence.

The logic behind simplification relies on knowing how factorials unfold, with each number descending by one till 1. In our example, simplifying the initial setup results in an easily understandable format: choosing one item from 20 options ends with only one valid sequence - picking the specific item itself, giving a result of 20.