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Understanding whether a number is even or odd is essential in combinatorial probability. Odd numbers are not divisible by 2, while even numbers are. When adding numbers, the sum of two odd numbers or two even numbers results in an even number. For example, 3 and 5 (both odd) sum to 8, an even number, while 4 and 6 (both even) also sum to 10, another even number. Conversely, combining one even and one odd number results in an odd sum. This characteristic is foundational when analyzing numerical combinations and their probabilities. Recognizing these arithmetic rules helps in determining combinations that yield an even number, which is the goal when calculating certain probabilities.

In probability problems, the use of combinations is crucial as it helps determine the different ways items can be selected. Combination formulas are used when the order of selecting items does not matter. For example, when choosing three numbers from a set of 100, the total number of combinations is calculated using the formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of selections. In this problem, the formula helps find the two scenarios leading to an even sum: selecting all three numbers as even or two odd numbers with one even. The combinations are calculated as follows: for three even numbers, it’s \(\binom{50}{3}\), and for two odd numbers with one even, it's \(\binom{50}{2} \times \binom{50}{1}\). These calculations show how many ways the favorable conditions can occur.

Probability calculation is the ratio of the number of favorable outcomes to the total number of possible outcomes. To find the probability that the sum of three selected numbers is even, we need to first determine the favorable outcomes using combinations. There are two favorable setups: all three chosen numbers are even, or two numbers are odd, and one is even. The computed combinations from these situations become our numerator. The denominator is the total possible combinations of selecting any three numbers from the set of 100, expressed as \(\binom{100}{3}\). Hence, the probability formula is: \[\frac{\left(\binom{50}{3} + \binom{50}{2} \times \binom{50}{1}\right)}{\binom{100}{3}}\]. This formula gives the likelihood of the sum being even, providing a clear path to find the desired probability.