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Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. It helps us figure out how to count different ways to choose items from a group. In the context of the Axline Computers problem, we're using combinatorics to figure out the number of ways to select certain employees from the total. This is where the combination formula, or "binomial coefficient," comes into play. The formula is
\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]
where \(n\) is the total number of items to choose from, \(k\) is the number of items to choose, and \(!\) denotes a factorial, which is the product of all positive integers up to that number.
In our scenario, when we're calculating \(\binom{40}{10}\), we're looking at how many ways we can choose 10 employees out of the 40 from Texas. Similarly, when using \(\binom{60}{10}\), we're considering all possible combinations of selecting 10 employees from the entire group of 60 employees.
With combinatorics, we can determine probabilities of different scenarios by comparing these various possible arrangements.

Random sampling is a fundamental concept used when selecting a subset from a larger population. This concept ensures that each member of the population has an equal chance of being chosen. It's a way to achieve unbiased representations in surveys, tests, or any situation where you want to make inferences from a smaller group.
In the Axline Computers example, we're tasked with randomly sampling 10 employees from a total of 60 employees at two locations. This randomness is crucial, as it guarantees an equal opportunity for selecting employees from either Texas or Hawaii. It also provides all possible combinations of employee groups without preferring one group over another.
Random sampling allows us to harness the power of probability theory to make generalizations about the larger group based on the sample. It is essential in statistical calculations as it forms the unbiased basis for analysis.

Statistical calculations involve using mathematical techniques to make conclusions or predictions based on data. In the context of probability, it’s about measuring the likelihood of various outcomes.
With the Axline Computers problem, we employed statistical calculations to determine the probability of different employee-group scenarios. By combing through different combinations of selecting employees and using the complement rule, we calculate probabilities for the specified criteria:

• 0 employees from Hawaii
• 1 employee from Hawaii
• 2 or more employees from Hawaii

For example, to find the probability of having zero employees from Hawaii, we calculate the ratio of ways to choose all employees from Texas to all possible ways to choose 10 employees from the total of 60. This is why we use the combination formula for both the numerator and denominator.
Complementing these detailed statistical calculations with adaptive understanding forms a robust foundation for making accurate predictions and decisions based on probable outcomes.