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Combinatorics is a branch of mathematics dealing with the study of counting, arrangement, and combination of objects. It gives us tools to count possible outcomes in a systematic way.
In problems like this one, combinatorics helps us determine all the different outcomes that could arise from different setups of an experiment.

• Each experimental setup has a fixed number of possible outcomes.
• The goal is to find all the different ways these setups can occur together.

In this problem, each setup acts independently of the others. This is important in combinatorial counting. Since all setups are separate, we need only to add their outcomes. While often, combinatorial problems multiply outcomes to find combinations, in this particular case, the task requires adding because we're interested in the total outcomes resulting from separate setups, not their combinations.

Probability is a way of thinking that helps us measure how likely something is to happen. When we talk about probability, we often refer to the likelihood of various outcomes.
Each setup in our experiment represents a chance or probability of a certain outcome happening.

• If you only knew the number of possible outcomes from each setup, you might think about what likelihood each outcome represents.
• Probability becomes important when comparing setups, or when weighing one setup's outcomes against another's.

In this exercise, if we were to assign probability, we could say each distinct outcome from all setups might be equally likely. Understanding how probability functions in an experiment can illuminate how decisions are made based on the potential frequency of certain outcomes.

Mathematical operations like addition, subtraction, multiplication, and division are tools that let us manipulate numbers in meaningful ways. In this problem, we see addition being used.
This is a basic but essential operation to tally total possible outcomes.

• Addition combines different sets of numbers into a single total.
• Understanding this operation's role helps break down complex problems into simple steps.

When adding outcomes from each setup, the operation is straightforward: we sum up the total number of outcomes available from all the items in the setups.
This approach ensures every potential path through the experiment is counted toward the final outcome. Understanding mathematical operations gives us the power to find solutions efficiently and confidently.