The big idea: When you expand , the numbers in front of the terms are row of Pascal's triangle. Each row opens and closes with 1, and every number inside is the two above it added together.
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Small n only: Pascal's triangle is quickest for a small (up to about 6).
For a larger power, use nCr (next section) instead of writing out every row.
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Coefficients without the triangle: For a larger power, each coefficient is a combination — work it out with the formula, or with the nCr button on the calculator.
First — what the ! sign means
- (say “n factorial”) means times every whole number below it down to 1 — for example .
- the total you're choosing from
- how many you choose from it
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The 15 you just found sits here in row 6.
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A bigger one
Find .
Step by step
- Substitute and , keeping three factors on top.
- Work it out.
Final answer
— far too big to reach by drawing the whole triangle.
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GDC tip (Paper 2): On the calculator: type , then MATH → ▶ (PRB) → 3: nCr, then .
So 8 nCr 3 = 56 here, and a big one like 12 nCr 3 = 220 is instant — far faster than drawing the triangle.
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Putting it together: The binomial theorem writes as a sum of terms , and there are exactly terms.
Read the structure
Write out the structure of .
Step by step
- Row 6 of the triangle gives the seven coefficients.
- The power of falls ; the power of rises .
Each number is the coefficient of that term — 7 numbers, so 7 terms.
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POWER OF a · a⁶ → a⁰
- The power of a counts down from 6 to 0.
- → a⁶, a⁵, …, a¹, a⁰
POWER OF b · b⁰ → b⁶
- The power of b counts up from 0 to 6.
- → b⁰, b¹, …, b⁵, b⁶
HOW MANY TERMS · 6 + 1 = 7
- One term for each coefficient, so there are 7 terms; in every term the a- and b-powers add to 6.
- → 7 terms in all
An exam-style question
In the expansion of , show that the coefficient of is .
Step by step
- Build row 5. For a small power you don't need the formula — the coefficients of are just row 5 of Pascal's triangle. Build it by adding neighbours in row 4.
- Read off . Row 5 lists the coefficients of in order, so the number in front of is the 3rd one.
Final answer
So the coefficient of in is — the 3rd number in row 5.
Row 5 — the 3rd number, in front of , is 10.
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To get full marks
- Write row 5 of Pascal's triangle, (add neighbours in row 4).
- Match it to the powers — the coefficient is the 3rd number.
- State that the coefficient of is 10.