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In statistics, a combination refers to a way of selecting items from a larger pool, where the order of selection does not matter. This concept is crucial when you want to know how many ways you can choose a subset of items from a set. The combination formula helps to determine the number of these possible selections. The formula is given by:

\[{_nC_r} = \frac{n!}{r!(n-r)!}\]

Here, \(_n \) represents the total number of items to choose from, and \(_r \) is the number of items to be chosen. The exclamation mark (!) indicates a factorial function. Understanding this formula allows you to solve problems like \({}_{48}C_{3}\), where you need to select 3 items from a pool of 48. Using the formula, you substitute 48 for \(_n\) and 3 for \(_r\).

Factorials are a key part of combinatorics and are denoted by an exclamation mark (\(!\)). A factorial of a non-negative integer \(_n\) is the product of all positive integers less than or equal to \(_n\). For example, \(_3! = 3 \times 2 \times 1 = 6\)

Factorials grow very quickly with larger numbers, which is why they are so useful in combinations and permutations. They help to simplify many combinatorial expressions by providing a way to manage large numbers and complex calculations. For instance, in our problem, the factorials are used as:

• \(48! = 48 \times 47 \times 46 \times 45!\)
• \(3! = 3 \times 2 \times 1 = 6 \)
• \(45! \) cancels out later in the calculation

By canceling out similar terms, the calculation simplifies significantly, making it much easier to handle.

Combinatorics is a field of mathematics focused on counting, arrangement, and combination of objects. It's an essential part of algebra and probability, often used in situations where you need to determine the possible ways objects can be arranged. The primary aim is to systematically count the number of possible outcomes. In problems involving combinations, you're often calculating how many ways you can choose a subset of items from a larger set, without worrying about the order.

Understanding combinatorics involves:

• Knowing the difference between combinations and permutations (order matters in permutations).
• Using the combination formula \(\binom{n}{r}\).
• Applying factorials to solve complex problems efficiently.

For the example \({}_{48}C_{3}\), combinatorics tells us we're interested in selecting 3 items out of a possible 48, and the steps to calculate this efficiently are crucial to understanding the number of ways this can happen. In essence, combinatorics is like solving a big puzzle, making it both challenging and fascinating.