Final answer:
The factorial notation indicates the number of ways to arrange a set number of items. For arranging 18 dots, none of the given options (2!, 3!, 4!, 5!) directly applies, implying that a more complex approach to permutations or combinations is needed.
Step-by-step explanation:
The question is asking for the different patterns to arrange 18 dots, and the options given are factorial notations. The factorial of a number n, denoted as n!, represents the product of all positive integers from 1 to n. In the context of arrangements or permutations, it indicates the number of distinct ways to arrange a given number of items. For instance, 3! would mean 3×2×1, which equals 6 possible arrangements. In this example, however, arranging 18 dots would likely require a factorial approach based on grouping or other rule-based approaches not directly represented by any of the given options (2!, 3!, 4!, 5!). Therefore, a more complex understanding of permutations and combinations would be required to determine how to arrange 18 dots, which could vastly exceed the quantities represented by the factorials listed.