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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 in combinatorial mathematics to determine the number of ways to choose a subset of a given size from a larger set. The binomial coefficient is denoted by the symbol "C(n, k)" or "n choose k," where "n" is the size of the larger set and "k" is the size of the subset.

The binomial coefficient can be calculated using the formula:

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 several important properties, including symmetry, which states that C(n, k) = C(n, n - k), and Pascal's identity, which states that C(n, k) = C(n - 1, k - 1) + C(n - 1, k). These properties make binomial coefficients useful in a variety of mathematical and computational applications, such as probability theory, statistics, and computer science.

In addition to their use in combinatorics, binomial coefficients also appear in other areas of mathematics, such as algebra and calculus. For example, they are used to expand expressions using the binomial theorem, which states that:

(x + y)^n = Σ [C(n, k) * x^(n - k) * y^k], for k = 0 to n

where Σ denotes the sum over all values of "k" from 0 to "n."

Overall, binomial coefficients are a fundamental concept in mathematics with a wide range of applications. Their properties and formulas make them a valuable tool for solving problems in combinatorics and other areas of mathematics.