Binomial theorem

The binomial theorem is an important formula about the expansion of powers of sums. It simplest version reads

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}$(1)

whenever n is any positive integer (but see below) and the numbers

${n \choose k}={\frac {n!}{k!(n-k)!}}$

are the binomial coefficients. This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was however known long before to Chinese mathematicians.

The cases n=2, n=3 and n=4 are the ones most commonly used:

(x + y)² = x² + 2xy + y²

(x + y)³ = x³ + 3x²y + 3xy² + y³

(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a ring as long as xy = yx.

Isaac Newton generalized the formula to non-integral and negative powers by considering an infinite series:

$(x+y)^{r}=\sum _{k=0}^{\infty }C(r,k)x^{k}y^{r-k}$ (2)

Here, r can be any real or complex number, and the sum will converge whenever the real or complex numbers x and y are "close together" in the sense that the absolute value |x/y| is less than one.

The geometric series is a special case of (2) where we chose y = 1 and r = -1.