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Discrete mathematics encompasses a wide range of topics including algorithms, graph theory, and combinatorics, which is highly applicable when solving problems involving countable, distinct values. When we look at finding the sum of even digits in three-digit numbers, we are actually applying one of discrete mathematics' subfields—combinatorics. Instead of being overwhelmed at the thought of verifying this condition for each number from 100 to 999 individually, discrete mathematics provides us with tools to analyze and categorize the numbers methodically.

For example, we use combinatorial reasoning by categorizing numbers into even or odd, and grouping them based on the sum of their digits. We determine the total count without listing them exhaustively—this is much more efficient and less prone to error than brute-force checking each number.

Number theory is a branch of pure mathematics devoted to the study of the integers and integer-valued functions. Within number theory, the concept of parity—which refers to whether an integer is even or odd—is crucial in solving our particular problem. By understanding that the sum of two even numbers or two odd numbers is always even, while the sum of an even and an odd number is always odd, we can predict the parity of a sum involving multiple digits.

In the context of the exercise, parity helps us to realize that we can only reach an even sum if we have an odd number of odd digits or no odd digits at all. This underlying principle is a result of number theory's fundamental properties and it's what guides our step of categorizing possible digit combinations into 'EEE' or 'OOE'.

Combinatorics, a field in discrete mathematics, relates to counting and arranging groups of objects according to certain rules. Here, it allows us to efficiently determine the number of three-digit numbers that have an even digit sum. We consider the positions where even and odd numbers can occur and calculate the total number of combinations.

For this exercise, combinatorics is used to calculate combinations of 'EEE' and 'OOE' without actually listing out all of the numbers. This method of using combinatorics for counting is much more powerful and generalizable than going through cases one by one, allowing us to solve the problem elegantly and systematically.

Parity is a mathematical term that expresses the attribute of an integer's evenness or oddness. It's a concept within number theory and is also closely related to combinatorics. In the context of our problem, we use parity to determine the nature of the digits' sum.

Through understanding parity, we assert that an even number of odd digits will give us an even sum. The advantage of this approach is its simplicity—knowing parity rules lets us quickly sort potential digit sums into 'even' or 'odd' without further calculation. Moreover, the parity principle is critical when reconciling our two different approaches to solve the problem: Knowing that not every two-digit combination will lead to a valid three-digit number, because of the hundreds place not including zero, we acknowledge that the combinatorial strategy provides more accuracy.