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Binomial coefficients, also known as binomial numbers, are a set of positive integers that occur as coefficients in the binomial theorem. They are used to represent the number of ways to choose k elements from a set of n elements without regard to the order of selection. The binomial coefficient is denoted by the symbol "n choose k" or C(n, k), and is defined mathematically as:

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

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

Binomial coefficients have a wide range of applications in mathematics, including combinatorics, probability, and algebra. They are used to calculate the number of combinations, to expand polynomials using the binomial theorem, and to solve problems involving probability and statistics.

The binomial coefficients can also be arranged in a triangular array known as Pascal's triangle, where each number is the sum of the two numbers directly above it. This pattern allows for the efficient calculation of binomial coefficients and has been studied extensively for its mathematical properties and applications.

In summary, binomial coefficients are a fundamental concept in mathematics with numerous applications in various fields. They provide a way to count combinations and expand polynomials, and their properties have been extensively studied and utilized in mathematical research.