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.
Press ▶ to build the triangle one row at a time — watch each inside number form as the sum of the two above it. Click any number to see how it is made, and how that row expands (a + b)ⁿ.
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IB-style question — expand with the triangle
Use Pascal's triangle to expand (a + b)³.
Step by step
- (a + b)³ just means three brackets multiplied together. Each term in the answer comes from picking one letter — a or b — out of every bracket.
- Every term uses all 3 brackets, so the powers of a and b always add up to 3. That is why a's power falls 3 → 0 as b's power rises 0 → 3 — they trade off.
- The number in front of each term comes straight from row 3 of Pascal's triangle — just read them off; you never have to work them out.
- Put the coefficients onto the terms.
Final answer
a³ + 3a²b + 3ab² + b³ — and the powers in every term add to 3.
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.
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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.
First — what ! and the brackets mean: Two symbols to read before the formula:
n! (say "n factorial") means multiply n by every whole number below it, down to 1 — e.g. 4! = 4 × 3 × 2 × 1 = 24.
The bracket ( n on top, r on the bottom ) is just another way of writing nCr — a number you work out from the formula (or the GDC's nCr button).
Press ▶ to read ⁵C₂ one line at a time — what ! means and how to work it out to 10.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
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 — find one coefficient
Find the coefficient of x³ in the expansion of (1 + x)⁴.
Step by step
- Use row 4 of Pascal's triangle — 1, 4, 6, 4, 1 — as the coefficients. The first term is 1, and 1 to any power is still 1, so those numbers sit straight in front of 1, x, x², x³, x⁴.
- Pick out the x³ term: it is 4x³, so the coefficient is 4.
- The 4 is exactly ⁴C₃ — and you read both numbers off the question: the top (4) is the power on the bracket, (1 + x)⁴, and the bottom (3) is the power of x you want, x³. So for a power too big to write out, just work out that one number.
Final answer
The coefficient of x³ is 4 (which is ⁴C₃). For a big power, get it straight from nCr instead of expanding.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
GDC tip (Paper 2): On the GDC: type n, then MATH → ▶ (PRB) → 3: nCr, then r.
So 4 nCr 3 = 4 here — and for a big one like 10 nCr 4 = 210, the GDC is far faster than drawing the whole triangle.
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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.