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.
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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 the ! sign means
- n! (say "n factorial") means multiply n by every whole number below it, down to 1 — e.g. 4! = 4 × 3 × 2 × 1 = 24.
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- the total you're choosing from
- how many you choose from it
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.
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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.
An exam-style question
Show that .
Step by step
- Put the numbers in. With and , write the choose formula before working anything out.
- Cancel, then work it out. The on the top and bottom cancel, leaving . Show this working, not just the answer.
Final answer
So , as required.
To get full marks
- Show the calculation: write and the cancel step .
- State the result: 120.
- On a show-that, writing only 120 with no steps can cost you — show the method that reaches the number.