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Combinatorics is the branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints, such as those concerning the order of the elements. In the context of a state lottery, combinatorics plays a crucial role in determining the number of possible outcomes when numbers are drawn.

For instance, in 'The Numbers Game', the lottery involves drawing four numbers, with each draw having 10 possible outcomes (digits from 0 to 9). Since each number is drawn with replacement, the same number can be drawn in more than one position. The combinatorial calculation for the total number of outcomes uses the rule of the product, calculating all possible combinations of the digits. This is computed as:
\begin{align*}10 \times 10 \times 10 \times 10 & = 10^4. \text{align*}Hence, there are 10,000 (or \(10^4\)) possible combinations when four numbers are drawn with replacement from a set of 10 digits.

Finite mathematics refers to areas of mathematics that deal with mathematical concepts and techniques which are typically used in real-world applications of business, industry, and other practical fields. It encompasses topics like algebra, graph theory, mathematical modeling, and especially probabilities, as seen in state lotteries.

When applying finite mathematics to 'The Numbers Game', it's important to understand that the lottery can be modeled using finite probabilistic methods. Since the game has a finite number of possible numerical outcomes, tools from finite math help to compute the probability of drawing a specific set of numbers. It is these finite properties that allow for a concrete calculation of winning odds, making the seemingly random lottery a system governed by mathematical principles.

Probability calculation is the process of determining the likelihood of a particular event occurring, often expressed as a fraction or percentage. The basic principle is to divide the number of ways the desired event can happen by the number of total possible outcomes.

In 'The Numbers Game', since there is only one grand prize-winning combination, the number of winning combinations is 1. As we learned from combinatorics that the total number of possible outcomes is \(10^4\), the probability of a ticket holder winning the grand prize is calculated by dividing the number of winning combinations (1) by the total number of possible outcomes (\(10^4\)). This gives:\begin{align*}\frac{1}{10^4} \end{align*} Therefore, the probability that a ticket holder has the winning ticket is \begin{align*}\frac{1}{10000}, or 0.01%. \end{align*} This calculation shows that the chances of winning the grand prize are quite low, which is typical for state lotteries.