The Binomial Theorem

Discrete Math · Axiom Academy

Understanding how (x+y)ⁿ expands using combinations and the power of counting arguments This beautiful result connects algebra (polynomial expansion) with combinatorics (counting problems). The coefficients aren't random—they count selections! Let's think about what happens when we expand (a+b)ⁿ. We're multiplying n copies of (a+b) together: To get a term with aᵏbⁿ⁻ᵏ, we need to choose a from exactly k of the n factors, and b from the remaining (n-k) factors. How many ways can we do this? Choose a from 2 of the 3 factors: C(3,2) = 3 ways Choose b from the remaining 1 factor Result: coefficient is 3, giving us 3a²b 3. Expanding (a+b)⁴ Step by Step Let's see the binomial theorem in action by expanding (a+b)⁴ term by term, with each coefficient explained combinatorially. Breaking down each coefficient: C(4,0) = 1: Choose a from 0 factors → 1 way → a⁰b⁴ = b⁴ C(4,1) = 4: Choose a from 1 factor → 4 ways → 4ab³ C(4,2) = 6: Choose a from 2 factors → 6 ways → 6a²b² C(4,3) = 4: Choose a from 3 factors → 4 ways → 4a³b C(4,4) = 1: Choose a from 4 factors → 1 way → a⁴ 4. Pascal's Triangle Connection The binomial coefficients form Pascal's Triangle, where each number is the sum of the two numbers above it. This reflects the combinatorial identity: Probability: Calculating binomial probabilities Computer Science: Analyzing algorithm complexity Number Theory: Proving divisibility properties Calculus: Deriving power rule and Taylor series

This is the written version of the interactive lesson above. See the full Discrete Math course.