Final answer:
The question presents disparate elements, primarily relating to combinatorics and algebra, but lacks cohesion, making it challenging to provide a definite answer. The concepts of combinations, algebraic simplifications, and the formula for calculating the total number of cases are embedded within the question. Additional context is needed to deliver a clear and precise solution.
Step-by-step explanation:
The question seems to have several parts that are disjointed, but it primarily appears to tackle concepts in combinatorics and algebra. When dealing with expressions for combinations and algebraic manipulations, certain formulas and algebraic techniques are utilized to simplify expressions and determine the total number of cases, combinations, or outcomes. It's important to first understand the basics of combinations, which are represented by the notation [n choose k], where n corresponds to the total number of items and k is the number of items chosen.
The reference information given mentions that k is the number of groups and n is the total number of cases. Specifically, n = 4(30) = 120, which suggests we are dealing with groups and possibly combinations of these groups. However, without a clear, single question to address, it's challenging to provide a concise answer without speculation. To address the question posed in a useful manner, additional clarity and context would be necessary. A key concept exemplified in the information provided is the algebraic manipulation to show that a sum of terms can be equivalent to n squared (n²), through smart addition and subtraction within a series.
The other references seem to discuss different topics such as reaction rates in chemistry and probabilities which are not directly related to the initial question about writing an expression as a single combination. Therefore, without further context, this collection of information cannot be explicitly solved or explained as a single problem. Nonetheless, this discussion highlights essential algebraic thinking skills that could be applied across multiple mathematical disciplines, including combinatorics and probability.