Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of elements within sets, often under specific constraints. It plays a crucial role in fields like computer science, probability, and optimization by providing foundational techniques for solving complex problems involving discrete structures.
Binomial coefficients are numerical factors that multiply the successive terms in the expansion of a binomial raised to a power, represented as 'n choose k' or C(n, k), and are calculated using the formula n! / (k!(n-k)!). They have applications in combinatorics, probability, and algebra, particularly in calculating combinations and understanding the structure of Pascal's Triangle.