The binomial theorem and Pascal's triangle
The binomial theorem expands (a + b)ⁿ without long multiplication. The coefficients come from Pascal's triangle, or from nCr.
Pascal's triangle, rows 0 to 6
| n | Coefficients | Sum (2ⁿ) |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 1 | 2 |
| 2 | 1 2 1 | 4 |
| 3 | 1 3 3 1 | 8 |
| 4 | 1 4 6 4 1 | 16 |
| 5 | 1 5 10 10 5 1 | 32 |
| 6 | 1 6 15 20 15 6 1 | 64 |
Formulas
How the triangle is built
- Each row starts and ends with 1.
- Every inside number is the sum of the two numbers just above it, e.g. 4 + 6 = 10 in row 5.
- Row n gives the coefficients of (a + b)ⁿ, so row 4 (1 4 6 4 1) is for power 4.
- Powers of a fall from n to 0 while powers of b rise from 0 to n.
Expand with Pascal's triangle
Expand (x + 2)⁴.
- Row 4 coefficients: 1, 4, 6, 4, 1. Here a = x and b = 2.
- 1 × x⁴ × 2⁰ = x⁴
- 4 × x³ × 2¹ = 8x³
- 6 × x² × 2² = 24x²
- 4 × x¹ × 2³ = 32x
- 1 × x⁰ × 2⁴ = 16
- Check with x = 1: (1 + 2)⁴ = 81 and 1 + 8 + 24 + 32 + 16 = 81. ✓
(x + 2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16
Find one term
Find the 4th term of (2x − 3)⁵.
- The 4th term is T(r+1) with r = 3. Here n = 5, a = 2x, b = −3.
- T4 = 5C3 (2x)⁵⁻³ (−3)³ = 10 × (2x)² × (−27).
- (2x)² = 4x², so T4 = 10 × 4x² × (−27) = −1,080x².
T4 = −1,080x²
A quick growth estimate
A tea exporter expects export prices to rise 2% a year for 5 years. Estimate the growth factor (1.02)⁵ using the first three binomial terms.
- (1 + x)ⁿ ≈ 1 + nx + nC2 x² with n = 5 and x = 0.02.
- nx = 5 × 0.02 = 0.1.
- nC2 x² = 10 × 0.0004 = 0.004.
- Sum: 1 + 0.1 + 0.004 = 1.104.
- A calculator gives 1.02⁵ = 1.1040808, so the estimate is very close.
About 1.104, i.e. prices about 10.4% higher after 5 years.
Compound growth (1 + r)ⁿ is a binomial. The first three terms, 1 + nr + nC2 r², let a Ceylon Tea exporter or a bank officer estimate a few years of growth in their head, then confirm on a calculator.
Short answer: “How many terms are there in (a + b)⁹?” n + 1 = 10 terms.
Structured question: “Expand (2x − 1)⁴ using Pascal's triangle.” Row 4: 1 4 6 4 1; signs alternate: 16x⁴ − 32x³ + 24x² − 8x + 1.
Structured question: “Find the coefficient of x² in (x + 3)⁴.” T3 = 4C2 x² 3² = 6 × 9 = 54.
Build Pascal's triangle by adding the two numbers above. In each term the powers add up to n, and with a minus sign the terms alternate.