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Factorials are mathematical expressions used extensively in permutations and combinations. A factorial is the product of all positive integers up to a given number. For example, the factorial of 5, denoted as 5!, is calculated by multiplying all integers from 1 to 5: \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials are particularly useful for finding the number of ways to arrange a set of items.
Permutations refer to the arrangement of objects in a specific sequence. When dealing with permutations, the order of arrangement matters. In this exercise, we see permutations at play since each of the three positions in the hospital must be filled by different individuals from the group of 15 candidates. The number of permutations of these candidates when choosing 3 people is expressed as \(15 \times 14 \times 13\), reflecting the need to choose one candidate for each position without repetition.

Combinatorics is the branch of mathematics dealing with counting, combinations, and permutations. It is applicable in situations where you want to determine the number of possible arrangements or selections of objects. Combinatorics simplifies complex counting problems by providing systematic approaches to tackle them.In our hospital scenario, the problem is a classic example of permutations because the order of filling the positions changes the outcome. By applying combinatorial principles, we recognize that the day nursing supervisor, night supervisor, and nursing coordinator must all be uniquely assigned to different candidates.
Unlike combinations where order doesn't matter (such as selecting a team of members), with permutations, the ranking or place in line is crucial. Thus, the calculation of \(15 \times 14 \times 13 = 2730\) directly results from the concept of permutations in combinatorics.

Problem solving in permutations often requires a structured approach. Here’s a simple guide to tackle such permutation problems:

• **Understanding the Scenario**: Begin by clearly identifying the positions or roles that need to be filled. Establish whether the order of selection is important.
• **Determine Possible Choices**: List the total number of options available initially, then sequentially reduce the options as choices are made, just like in our hospital exercise.
• **Formulating the Equation**: Set up a multiplication sequence that reflects the selection process, decreasing the number of available candidates with each choice made.
• **Performing Calculations**: Execute the multiplication to find the total number of permutations. In the case of our example, calculate \(15 \times 14 \times 13\) to determine the number of ways to fill the positions.
• **Verification and Conclusion**: Review the calculations for accuracy and consider if the solution logically aligns with the problem's conditions.

By following these steps, we ensure a robust approach to solving complex permutation challenges, minimizing errors and enhancing confidence in the solution.