Introduction
Pascal’s Triangle is one of the most famous structures in mathematics, appearing in algebra, probability, and combinatorics. For IB Math students in both HL and SL, Pascal’s Triangle is most directly linked to the binomial theorem, where it provides coefficients for expansions.
Beyond expansions, Pascal’s Triangle reveals fascinating number patterns, symmetry, and connections to probability—making it a powerful study tool for IB students.
Quick Start Checklist
To master Pascal’s Triangle in IB Math:
- Learn how to construct it row by row.
- Connect triangle rows to binomial expansions.
- Use it to find binomial coefficients without calculating factorials.
- Explore its symmetry and patterns.
- Practice IB-style questions involving expansions and probability.
Constructing Pascal’s Triangle
Pascal’s Triangle starts with 1 at the top. Each number inside the triangle is the sum of the two numbers above it.
Example (first six rows):
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Pascal’s Triangle and the Binomial Theorem
Each row of Pascal’s Triangle gives the coefficients for the expansion of (a + b)ⁿ.
- Row 0: 1 → (a + b)⁰ = 1
- Row 1: 1 1 → (a + b)¹ = a + b
- Row 2: 1 2 1 → (a + b)² = a² + 2ab + b²
- Row 3: 1 3 3 1 → (a + b)³ = a³ + 3a²b + 3ab² + b³
This avoids having to calculate nCk each time.
