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Combinatorics is a fundamental area in mathematics focused on counting, arrangements, and combinations. It is particularly useful when we need to determine the number of ways something can occur. In our scenario, combinatorics helps us calculate the number of ways to select a group of workers from different shifts.
To determine how many ways we can choose workers from a certain shift, the concept of combinations is used. The formula for combinations is \(inom{n}{k}\), which represents the number of ways to choose \(k\) items from \(n\) total items without regard to order.
In the example given, if we have 30 workers on the day shift and need to select 9 of them, the number of ways to do this is \(\binom{30}{9}\). Here, \(n = 30\) and \(k = 9\).
This approach helps us understand various selection-related problems without having to manually list all possibilities, making problem-solving efficient.

Probability calculation is the process of quantifying the likelihood of an event happening. It is expressed as a number between 0 and 1. A probability of 0 means the event will not occur, while a probability of 1 means it will certainly occur.
In our problem, we calculate probabilities by dividing the number of successful outcomes by the total number of possible outcomes. For instance, to find the probability that all 9 workers selected are from the day shift, we use the formula:
\[P(\text{all from day shift}) = \frac{\binom{30}{9}}{\binom{67}{9}}\]
This expression represents the ratio of the number of ways to choose 9 workers out of the 30 day shift workers to the number of ways to choose any 9 workers from all 67 workers in the factory.
If you're calculating the probability that all 9 workers come from the same shift, you sum the probabilities for each shift:

• Day shift probability
• Evening shift probability
• Morning shift probability

Adding these gives the total probability that all 9 come from the same shift.

Shift work is a working system where different groups of workers perform tasks during different times of the day. This is common in industries that require 24-hour operation, like manufacturing or healthcare. Each shift has its specific start and end times.
In our example, the day shift runs from 08:00 to 16:00, the evening shift from 16:00 to 24:00, and the morning shift from 00:00 to 08:00. This arrangement allows the factory to maintain continuous productivity.
By understanding shift work, we can better grasp probability concepts within this structure. Knowing who is in each shift allows us to categorize and calculate probabilities more effectively. It also becomes clear how each shift's characteristics might impact the selection of workers, such as the differing numbers of employees per shift, which in turn affects probability outcomes.