The Binomial Theorem
Simple Explanation
The Binomial Theorem gives the complete expansion of (a+b)ⁿ for ANY power n, all at once: (a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ, summed over r = 0 to n. It generalizes Pascal's-triangle expansion to work for any n, even very large ones, without needing to build the whole triangle first.
Why Do We Need It?
This is the general, formula-based version of what Pascal's triangle does visually — it works for any power n, including large values where building a triangle by hand would be impractical.
Formula
The Binomial Theorem
(a + b)ⁿ = Σ (from r=0 to n) of ⁿCᵣ · aⁿ⁻ʳ · bʳ
Gives every term of the expansion of (a + b)ⁿ directly, without multiplying out the brackets by hand — each term's coefficient is a binomial coefficient, and the exponents of a and b always add to n.
- n
- — the power the binomial is raised to
- r
- — the term index, running from 0 to n
- ⁿCᵣ
- — the binomial coefficient for that term
When to use it: Whenever you need to expand (a + b)ⁿ for any n, especially when n is too large to expand by repeated multiplication.
Worked Example
Apply the binomial theorem directly
Use the binomial theorem to find the first three terms of (x + 3)⁶.
Why Does This Work?
Stated simply: (a+b)ⁿ means multiplying n copies of (a+b) together. Expanding this product means picking either a or b from each of the n factors, in every possible combination, and adding up all the results. Collecting together every way of picking exactly r copies of b (and n−r copies of a) gives the term aⁿ⁻ʳbʳ, and there are exactly ⁿCᵣ such ways (choosing which r of the n factors contribute a b) — summing over every possible r from 0 to n accounts for every term, proving the theorem.
Real-Life Example
Probability of a fixed number of successes
The probability of getting exactly r successes in n independent trials, each with fixed success probability, uses binomial coefficients in exactly this pattern.
The binomial probability formula is built directly on the binomial theorem's ⁿCᵣ structure — the same coefficients doing double duty in algebra and probability.
Practice
Find the coefficient of the x³ term (i.e. r=1) in the expansion of (x + 2)⁴ using the binomial theorem. (This is the ⁴C₁ · 2¹ part.)
HardCommon mistake
Forgetting that the exponents of a and b must always sum to n in every term — a term like a³b³ from an expansion of (a+b)⁵ would be an error, since 3+3=6, not 5.
Quick Review
- (a+b)ⁿ = Σ (r=0 to n) ⁿCᵣ aⁿ⁻ʳ bʳ.
- Works for any power n, unlike reading a specific row of Pascal's triangle by hand.
- Proof idea: counts how many ways each aⁿ⁻ʳbʳ term can arise from multiplying out n factors of (a+b).