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 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.