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Combinations, in mathematics, refer to the selection of items from a larger set where the order of selection does not matter. It is one of the fundamental concepts in combinatorics, a branch of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.

To understand combinations, let’s start with a simple example. Suppose you have a set of three items: {A, B, C}. How many ways can you select two items from this set? You could have AB, AC, or BC. Notice that BA, CA, and CB are not counted as separate combinations because, in combinations, the order of selection does not matter. So, there are three combinations of two items from a set of three.

The mathematical formula for calculating combinations is given by the binomial coefficient. If you have a set of n items and you want to know how many ways you can select r items, the formula is:

C(n, r) = n! / [r!(n-r)!]

Here, ‘!’ denotes factorial, which is the product of all positive integers up to that number. For example, 5! = 5*4*3*2*1 = 120.

One of the most common applications of combinations is in probability theory. If you’re trying to calculate the probability of a certain event happening, you often need to know how many possible outcomes there are, and combinations can help you determine this.

For instance, if you’re dealt a hand of 5 cards from a standard deck of 52, how many possible hands are there? Using the formula for combinations, the answer is C(52, 5) = 52! / [5!(52-5)!] = 2,598,960 possible hands.

It’s important to distinguish combinations from permutations. While combinations are concerned with the selection of items where the order does not matter, permutations are all about the arrangement of items where the order does matter.

In summary, combinations are a fundamental concept in mathematics that allow us to calculate the number of ways to select items from a larger set. They are particularly useful in probability theory, where they can help us determine the number of possible outcomes for a given event.

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