Binomial theorem

The binomial theorem expresses a nonnegative integer power of a sum as a finite expansion whose terms use binomial coefficients and powers of the summands.

Binomial coefficient: A number counting ways to choose a fixed-size subset from a finite set. Each expansion term is weighted by the number of ways to choose which factors contribute the second summand.
Binomial: An algebraic expression consisting of two terms joined by addition or subtraction. The theorem applies specifically to powers of expressions with two terms.
Polynomial multiplication: The operation of multiplying polynomials by combining products of their terms. The theorem shortcuts repeated multiplication of identical binomials.
Al-Karaji: A tenth-century Persian mathematician known for early systematic work on algebra and polynomial operations. His work includes an early account of a rule equivalent to the binomial expansion.
Generalized binomial theorem: An extension of the binomial expansion to exponents that need not be nonnegative integers, often yielding an infinite series. It extends the exponent range but may replace the finite sum with an infinite series.
Pascal's triangle: A triangular arrangement of numbers in which each interior entry is the sum of the two entries above it. Its rows list the coefficients in successive binomial expansions.
Exponentiation: The operation of raising a number or expression to a power. A nonnegative integer exponent means repeated multiplication of the binomial.
Binomial distribution: The probability distribution for the number of successes in a fixed number of independent trials with equal success probability. Its probability formula uses binomial coefficients to count success patterns.
Omar Khayyam: A Persian mathematician, astronomer, and poet who studied algebraic equations and geometric methods. He used geometric reasoning related to binomial coefficients in solving higher-degree equations.
Geometric series: A series whose consecutive terms have a constant ratio. It is another power expansion, but its terms follow a ratio pattern rather than binomial coefficients.

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