Which theorem provides the expansion formula for (a+b)^n with integer n?

• A. Binomial Theorem
• B. Conic Section
• C. Theorem
• D. Pascal's Triangle

Answer: (A)

Expanding a power of a sum follows a rule that gives the exact coefficients in front of each term after you multiply out (a + b) n times. This rule is the Binomial Theorem. It says that for an integer n ≥ 0, (a + b) to the n equals the sum from k = 0 to n of binomial(n, k) times a^(n−k) times b^k. The binomial coefficients binomial(n, k) tell you how many different ways you can pick k factors of b from the n factors of (a + b), which aligns with the pattern you see in small n: (a + b)^2 = a^2 + 2ab + b^2, (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, and so on. Pascal’s Triangle is a handy visual aid for these coefficients, reading off the row n for the binomial coefficients, but the expansion formula itself is given by the Binomial Theorem.