Binomial Theorem

AKA: Binomial Expansion.
See: Elementary Algebra, Exponentiation, Binomial (Polynomial), Summation, Binomial Coefficient, Pascal's Triangle, Combinations.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Binomial_theorem Retrieved:2015-11-18.
- In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power $(x + y) n$ into a sum involving terms of the form $a x b y c$ , where the exponents $b$ and $c$ are nonnegative integers with $b + c n$ , and the coefficient $a$ of each term is a specific positive integer depending on $n$ and $b$ . For example, : [math]\displaystyle{ (x+y)^4 \;=\; x^4 \,+\, 4 x^3y \,+\, 6 x^2 y^2 \,+\, 4 x y^3 \,+\, y^4. }[/math] The coefficient $a$ in the term of $a x b y c$ is known as the binomial coefficient [math]\displaystyle{ \tbinom nb }[/math] or [math]\displaystyle{ \tbinom nc }[/math] (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also arise in combinatorics, where [math]\displaystyle{ \tbinom nb }[/math] gives the number of different combinations of b elements that can be chosen from an n-element set.

https://www.math.rutgers.edu/~cherlin/History/Papers1999/weiss.html
- QUOTE: Binomial Theorem: This is the usual expansion of (a+b)^n as a sum of monomials with certain standard coefficients. For n an integer, the expansion is finite and the coefficients are integers (occurring in Pascal's triangle). For other values of n, the formula becomes an infinite series and the coefficients are no longer integers, but are given by the usual formulas. This discovery was also one of the first made by Newton as a student.