combinations

Combinations are a fundamental concept in combinatorics, a branch of mathematics that deals with counting, arrangement, and combination of objects. A combination is a selection of items from a larger set, where the order of selection does not matter. This is in contrast to permutations, where the order of selection is significant.

The number of combinations of n items taken r at a time is denoted by the binomial coefficient, often written as "n choose r" or C(n, r). It is calculated using the formula:

C(n, r) = n! / (r! * (n - r)!)

where "n!" denotes the factorial of n, which is the product of all positive integers up to n.

For example, if we have a set of 5 items and we want to choose 3, the number of combinations is:

C(5, 3) = 5! / (3! * (5 - 3)!) = 10

This means there are 10 different ways to choose 3 items from a set of 5, without regard to the order of selection.

Combinations have numerous applications in various fields, including probability, statistics, computer science, and engineering. They are used to model and solve problems involving selection and counting, such as determining the number of possible hands in a poker game or the number of ways to arrange items in a lineup.