(a+b)² = a²+2ab+b², (a+b)³ = a³+3a²b+3ab²+b³ … If you have ever wondered what rule determines the coefficients each time you expand an expression, that rule is the binomial theorem. Change the exponent n with the slider to see how the expansion grows, and check which row of Pascal’s triangle matches the coefficients.
Binomial theorem formula: When you expand (a+b)ⁿ, each term has the form ₙCₖ · aⁿ⁻ᵏ · bᵏ (k = 0, 1, 2, …, n). Here, ₙCₖ is the "number of ways to choose k items from n items without regard to order," and it exactly matches the coefficient in the expansion.
Why does the number of combinations become the coefficient? When you expand (a+b)ⁿ = (a+b)(a+b)⋯(a+b) (multiplying n factors), each term comes from choosing b from some of the n parentheses and a from the rest. The number of ways to choose exactly k b’s is ₙCₖ, and that determines how many times the term aⁿ⁻ᵏbᵏ appears.