Binomial Theorem
Reference for the binomial theorem (a+b)^n expansion using binomial coefficients C(n,k).
Covers Pascal's triangle, algebra, probability, and combinatorics.
The Formula
(a + b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ for k = 0 to n
The binomial theorem expands any power of a two-term expression into a sum of terms. Each term involves a binomial coefficient C(n,k) multiplied by powers of a and b.
Variables
| Symbol | Meaning |
|---|---|
| a, b | The two terms of the binomial |
| n | The power to which the binomial is raised |
| k | Index of the current term (0, 1, 2, ..., n) |
| C(n,k) | Binomial coefficient = n! / (k!(n-k)!) |
Example 1
Expand (x + 2)⁴
C(4,0)x⁴(2)⁰ + C(4,1)x³(2)¹ + C(4,2)x²(2)² + C(4,3)x¹(2)³ + C(4,4)x⁰(2)⁴
= 1·x⁴ + 4·2x³ + 6·4x² + 4·8x + 1·16
= x⁴ + 8x³ + 24x² + 32x + 16
Example 2
Find the 3rd term of (3a - b)⁵
Third term: k = 2 (since k starts at 0)
C(5,2) × (3a)³ × (-b)² = 10 × 27a³ × b²
= 270a³b²
When to Use It
Use the binomial theorem when:
- Expanding expressions like (x + y)ⁿ without multiplying repeatedly
- Finding a specific term in a binomial expansion
- Working with probability distributions (binomial distribution)
- Simplifying algebraic expressions in calculus and physics
Key Notes
- The expansion of (a + b)ⁿ always produces exactly n + 1 terms, and the coefficients are symmetric — C(n,k) = C(n, n−k) — so the first and last coefficients are always 1
- Pascal's triangle gives the binomial coefficients directly: each row n gives the coefficients for (a+b)ⁿ; each entry equals the sum of the two entries directly above it, avoiding factorial computation for small n
- When b is negative, the signs of terms alternate: (a − b)ⁿ produces +, −, +, − … — a missed negative sign on an odd-powered b term is the most common expansion error
- The binomial approximation (1+x)ⁿ ≈ 1 + nx for |x| ≪ 1 is used throughout physics and engineering — for example, (1.001)¹⁰⁰ ≈ 1.1, and relativistic corrections at low speeds use this form