The big idea: The coefficients of (a + b)ⁿ are row n of Pascal's triangle — start each row with 1, and every inside number is the sum of the two above it.
Building the triangle
- Row 0: 1
- Row 1: 1 1
- Row 2: 1 2 1 (2 = 1 + 1)
- Row 3: 1 3 3 1 (3 = 1 + 2)
- Row 4: 1 4 6 4 1 (6 = 3 + 3)
IB-style question — expand with the triangle
Use Pascal's triangle to expand (a + b)³.
Step by step
- Row 3 of the triangle gives the coefficients.
- Powers of a fall 3 → 0; powers of b rise 0 → 3.
Final answer
a³ + 3a²b + 3ab² + b³.
Small n only: Pascal's triangle is quickest for small n (up to about 6). For a bigger power, use ⁿCᵣ (next section) instead of writing out every row.
Coefficients without the triangle: For bigger powers, a coefficient is a combination ⁿCᵣ — compute it with the formula, or with the GDC's nCr button.
IB-style question — by hand
Find ⁵C₂.
Step by step
- Substitute n = 5, r = 2.
- Cancel the 3! and compute.
Final answer
⁵C₂ = 10 (matches row 5: 1, 5, 10, 10, 5, 1).
IB-style question — a bigger one
Find ¹⁰C₄.
Step by step
- Substitute n = 10, r = 4; keep four factors on top.
- Compute.
Final answer
¹⁰C₄ = 210.
IB-style question — read a coefficient straight off
Find the coefficient of x³ in the expansion of (1 + x)⁴.
Step by step
- Every term of (1 + x)⁴ is a ⁿCᵣ times a power of x, so the coefficient of xʳ is just ⁴Cᵣ. The x³ coefficient is therefore ⁴C₃.
- Work out ⁴C₃ with the formula (or the GDC's nCr).
Final answer
The coefficient of x³ is ⁴C₃ = 4 — you never have to expand the whole bracket.
GDC tip (Paper 2): On the GDC: type n, then MATH → ▶ (PRB) → 3: nCr, then r.
So 10 nCr 4 = 210 — much faster than the triangle for large n.
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Putting it together: The binomial theorem writes (a + b)ⁿ as a sum of terms ⁿCᵣ aⁿ⁻ʳ bʳ — with exactly n + 1 terms.
IB-style question — read the structure
Write out the structure of (a + b)⁵.
Step by step
- Row 5 coefficients (six of them).
- Powers of a fall 5 → 0; powers of b rise 0 → 5.
Final answer
Six terms (n + 1 = 5 + 1).
How many terms?: (a + b)ⁿ always has n + 1 terms.
The power of a counts down from n to 0; the power of b counts up from 0 to n; in every term the two powers sum to n.