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When we talk about factorial calculation in mathematics, it refers to the product of all positive integers starting from a specific number down to one. This is represented by the symbol \textbf{!}. For instance, the factorial of 6, written as 6!, means multiplying 6 by every positive integer below it:
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \]
Just from this product, we can see that factorials grow extremely fast with larger numbers. This concept is pivotal in various areas of mathematics, including probability, combinations, and permutations. Recognizing when to apply factorial calculation often simplifies complex problems, enabling us to reach solutions with structured and comprehensible steps. Factorials are foundational, especially when dealing with permutations without repetition—as we will explore further.

Permutations are arrangements of objects where the order is important; and when we mention 'without repetition,' it indicates that each element is unique and can only be chosen once. The number of such permutations of n distinct objects is given by n!. As in our exercise example, ranking six distinct jokes from best to worse is a classic case of permutations without repetition.
Thus, the number of different ways to rank these jokes is simply worked out by calculating the factorial of 6 (6!). It's crucial to understand the 'no repetition' part because it affects how we calculate permutations. If repetition was allowed, the calculation would be different, leading to a much greater number of permutations. The logic behind this is that once each joke has taken a particular position in the ranking, it can't be used again for another position—hence the total number of arrangements is finite and governed by the no-repetition rule.

Understanding mathematical reasoning is key to solving any mathematical problem. It allows us to follow a sequence of logical steps to reach a conclusion or solve a problem. In the context of our textbook exercise, mathematical reasoning guided us to identify the problem type as a permutation without repetition.
Using deductive reasoning, we can see that because each joke must be ranked uniquely without the possibility of repeating positions, we apply the logic that this problem requires factorial calculation. Step by step, we deduced that to find the solution, one must calculate the factorial of the total number of unique items (jokes, in this case), represented by 6!. Such problem-solving skills are central to the study of mathematics and are transferable to numerous scenarios, enabling students to approach complex tasks systematically and with confidence.