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Combinatorics is a branch of mathematics that deals with counting combinations and permutations. The combination formula is a fundamental concept within this field. It helps us determine the number of ways to select a group of items from a larger set, where the order of selection doesn't matter.

In our exercise, we want to determine how many ways we can choose 10 nurses from a group of 90. The combination formula is represented as:\[C(n, r) = \frac{n!}{r!(n-r)!}\]

• \(n\) is the total number of items to choose from.
• \(r\) is the number of items we want to select.

The exclamation mark \(!\) denotes a factorial, a concept we'll explore next.

The beauty of the combination formula is that it simplifies the process of counting large sets efficiently without having to list out every possible combination manually. It's a powerful tool in probabilistic scenarios and can be applied in various fields like statistics, gambling, and computer science.

Factorials are quite a handy mathematical concept, particularly when dealing with combinations and permutations. A factorial, denoted by the symbol \(!\), is the product of all positive integers up to a given number.

For instance:

• The factorial of 5, written as \(5!\), is \(5 \times 4 \times 3 \times 2 \times 1 = 120\).
• \(0!\) is defined as 1, which is a key point to remember.

In our nurse selection example, we needed to compute a few factorials: \(90!\), \(10!\), and \(80!\).

Factorials grow very fast, which can make the computation tricky without the help of calculators or software. However, for combination calculations, many terms cancel out, simplifying our work often to manageable numbers. Understanding factorials is essential for using the combination formula correctly and interpreting the implications of large numbers.

Sample selection, as derived through the combination process, is about choosing items, like nurses, in contexts where the order does not matter. This is opposed to permutations, where order is crucial.

When selecting samples using the combination formula, our focus shifts solely to the choice of items irrespective of sequence. This approach suits our exercise perfectly where 10 nurses are to be chosen out of 90 without caring about the order in which they are picked.

The purpose of selecting samples varies based on field or scenario. For example:

• In healthcare studies, like ours, samples help in determining trends or outcomes by studying a smaller, manageable group.
• In surveys, samples reflect a larger population's opinions or conditions without needing input from everyone.

Thus, understanding how to select samples properly and use combinations enables us to ensure appropriate and effective data collection and representation.