Proving the Binomial Theorem

Combinatorics · Axiom Academy

LESSON Proving the Binomial Theorem Understanding why binomial coefficients appear when expanding (x + y)^n 1. The Combinatorial Foundation When we expand (x + y)^n, we're really multiplying n copies of (x + y) together: Each term in the expansion comes from choosing either x or y from each of the n factors. To get a term x^k y^(n-k), we must choose x from exactly k of the factors, and y from the remaining (n-k) factors. 2. Visualizing the Combinatorial Argument Let's focus on a specific example: expanding (x + y)^4 and finding the coefficient of x^2 y^2. We have 4 factors: (x + y)(x + y)(x + y)(x + y). To get x^2 y^2, we need to choose exactly 2 factors to contribute x, and the other 2 will contribute y. Each different selection of which 2 factors give us x produces the same term x^2 y^2. Since there are C(4,2) = 6 ways to choose 2 positions out of 4, the coefficient is 6. 3. The General Pattern and C(n,k) The pattern generalizes perfectly. For any term x^k y^(n-k) in the expansion of (x + y)^n: We have n factors of (x + y) to work with We need to choose exactly k of them to contribute x The remaining (n-k) factors automatically contribute y The number of ways to make this choice is C(n,k) This is our combinatorial proof : by counting the number of ways to form each term, we've proven the Binomial Theorem! 4. Proof by Induction (Outline) An alternative approach uses mathematical induction:

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