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Discrete mathematics is a branch of mathematics that deals with structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying smoothly, discrete structures such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values.

When we apply discrete mathematics to the problem of rolling two dice, we are counting the distinct possible outcomes of this activity. Each die has six faces, and each face corresponds to an integer value between 1 and 6, which is a finite, countable set. The exercise presented involves counting specific outcomes from a discrete set, which in this case is the product of two separate sets of die faces.

Probability theory is a branch of mathematics that deals with the analysis of random events. The main object of probability theory is the probability, a measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (inclusive), where 0 indicates an impossibility and 1 indicates certainty.

In our dice problem, we are essentially calculating probabilities by counting the number of favorable outcomes against the total number of possible outcomes. The solution involves finding all the ways at least one '2' can occur when rolling both dice. This counting method directly correlates with the concept of classical probability, where the probability is defined as the ratio of the number of favorable outcomes to the number of all possible outcomes.

Combinatorics is a field in discrete mathematics concerned with the study of combinations, arrangements, and permutations of a set of items. The branch has strong ties to probability theory as it supplies the underlying theory for counting the possible outcomes, which is fundamental in calculating probabilities.

In the context of our dice rolling exercise, combinatortics helps us understand the different ways that outcomes can be arranged when one or both dice show a '2'. The method used in the provided solution follows a combinatorial reasoning — first calculating the total number of combinations available when two dice are rolled, and then subtracting the combinations that do not meet the criteria (at least one die showing a '2'). This subtraction method avoids the need to list every single combination that meets the criteria, which is a common combinatorial technique for problem-solving.