Binomial coefficients: what are they?

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Binomial coefficients define the number of possible combinations when selecting 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 exclusively of binomial coefficients.

Mathematically, binomial coefficients are written as two vertically aligned numbers within a set of parentheses. The higher number, represented by “n”, is the total number of possibilities. Usually represented by “r” or “k”, the lower number is the number of unordered results to select from “n”. Both numbers are positive and “n” is greater than or equal to “r”.

The binomial coefficient, or the number of ways “r” can be extracted from “n”, is calculated using factorials. A factorial is a number multiplied by the next smallest number by the next smallest number, and so on until the formula reaches one. It is represented mathematically as n! = n(n – 1)(n – 2)…(1). The zero factorial equals 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. For example, if n is 5 and r is 2, 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 from the fraction. This results in (5*4)/(2*1), which is equal to 10.

The binomial theorem is a way to compute 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 successive term of the expansion is computed by adding 1 to the lower number of 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 lower number of the coefficient equals nf.

Each number in Pascal’s triangle is a binomial coefficient which can be calculated using the formula for binomial coefficients. The triangle starts with a 1 at the top, and each number in a row below can be calculated by adding 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 figurative numbers.

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