The binomial theorem states: (x + y)^n = $sum_{r=0}^{n}$ C(n,r) * $x^{n-r}$ * $y^r$ for positive integer n.

Key observations: (1) There are exactly (n+1) terms. (2) The powers of x decrease from n to 0, while powers of y increase from 0 to n. (3) In every term, the sum of exponents of x and y equals n. (4) C(n,r) = C(n, n-r), so the coefficients are symmetric about the middle.

Example: (a + b)^4 = $a^4$ + 4$a^{3b}$ + 6a^{2b}^2 + 4$ab^3$ + $b^4$.