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Combinatorics is a fascinating area of mathematics that deals with counting, arranging, and selecting objects. It is crucial when you need to answer questions about the number of ways things can be combined or organized. In the context of our problem, combinatorics helps us determine how many different groups of employees can be formed. We use combinations when the order of selection does not matter, which is exactly what we need here.

For example, to find ways to select 10 employees out of 60, we use the combination formula, represented as \( \binom{60}{10} \). This is pronounced '60 choose 10' and calculated as:

• First, determine the factorial of the total number of items (60!), which means multiplying all whole numbers from 60 down to 1.
• Next, determine the factorial of the number of chosen items (10!).
• Finally, compute the factorial of the difference of the two numbers \((60-10)! = 50!\).

The formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) gives us the number of combinations, where \( n \) is the total number of items, and \( k \) is the number of items to choose. This technique is used repeatedly throughout our probability calculation examples.

Sampling is the process of selecting a subset of individuals from a larger population to make conclusions about the whole. In the exercise problem, sampling is accomplished by randomly selecting 10 employees from a total of 60 employees in two different plants.

When sampling, it’s essential to understand the characteristics of the population to ensure the sample accurately represents it. This helps us compute probabilities reliably, as shown by determining how likely certain selections are. Sampling can be random, systematic, or stratified, but in our case, it's simple random sampling, meaning each employee has an equal chance of being picked.

Given two groups within the population (Texas and Hawaii plants), sampling questions often involve calculating probabilities regarding how many selections belong to each group. Here we've calculated the probability of selecting varying numbers of employees from each plant. The use of combinatorics makes determining these probabilities feasible and systematic.

The binomial coefficient is a numerical factor that arises in binomial expansions and is widely used in combinatorics. It represents the number of ways to choose k elements from a set of n elements without regard to the order of selection. This coefficient is denoted by \( \binom{n}{k} \).

In this exercise, the binomial coefficient is used to calculate the number of possible ways to select employees from each plant. For instance, to find how many ways you can select all 10 employees from the 40 in Texas, you calculate \( \binom{40}{10} \). Similarly, for one from Hawaii and nine from Texas, you'd compute \( \binom{20}{1} \times \binom{40}{9} \).

The binomial coefficient helps simplify complex problems, breaking them down into manageable calculations. When paired with the probability formula, it aids in figuring out the likelihood of different sampling scenarios, like what proportion of your sample comes from a specific site.