Binomial coefficients define the number of combinations that are possible when picking a certain number of outcomes from a set of a given size. They are used in the binomial theorem, which is a method of expanding a binomial — a polynomial function containing two terms. Pascal's triangle, for example, is composed solely of binomial coefficients.

Mathematically, binomial coefficients are written as two numbers vertically aligned within a set of parentheses. The top number, represented by "n," is the total number of possibilities. Usually represented by "r" or "k," the bottom number is the number of unordered outcomes to be selected from "n." Both numbers are positive, and "n" is greater than or equal to "r."

The binomial coefficient, or the number of ways that "r" can be picked from "n," is calculated using factorials. A factorial is a number times the next smallest number times the next smallest number, and so on until the formula reaches one. It is represented mathematically as n! = n(n - 1)(n - 2)...(1). Zero factorial is equal to one.

For a binomial coefficient, the formula is n factorial (n!) divided by the product of (n - r)! times r!, which can usually be reduced. If n is 5 and r is 2, for example, the formula is 5!/(5 - 2)!2! = (5*4*3*2*1)/((3*2*1)*(2*1)). In this case, 3*2*1 is in both the numerator and denominator, so it can be canceled out of the fraction. This results in (5*4)/(2*1), which equals 10.

The binomial theorem is a way to calculate the expansion of a binomial function, represented by (a + b)^n — a plus b to the nth power; a and b can be composed of variables, constants, or both. To expand the binomial, the first term in the expansion is the binomial coefficient of n and 0 times a^n. The second term is the binomial coefficient of n and 1 times a^(n-1)b. Each subsequent term of the expansion is calculated by adding 1 to the bottom number in the binomial coefficient, raising a to the power of n minus that number, and raising b to the power of that number, continuing until the bottom number of the coefficient equals n.

Each number in Pascal's triangle is a binomial coefficient that can be calculated using the formula for binomial coefficients. The triangle begins with a 1 at the top point, and each number in a lower row can be calculated by adding together the two entries diagonally above it. Pascal's triangle has several unique mathematical properties — in addition to binomial coefficients, it also contains Fibonacci numbers and figurate numbers.