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Pascal rows and binomial coefficient structure

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Which row rule should be used for the binomial power shown?

\[\left(x+y\right)^5\]
Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

Binomial coefficient structure

The binomial power determines the Pascal row and the coefficient pattern. Factor powers move in opposite directions, and subtraction produces alternating signs

\[\binom{n}{r}=\frac{n!}{r!\left(n-r\right)!}\]
  1. Match the row index to the power The power n selects Pascal row P_n for a binomial expansion \(\displaystyle \left(x+y\right)^4\longrightarrow P_4\)
  2. Add adjacent parents Each interior entry is the sum of the two entries directly above it \(\displaystyle P_4=\left(1,4,6,4,1\right)\)
  3. Apply alternating signs A subtraction in the binomial makes the terms alternate in sign \(\displaystyle \left(x-y\right)^3=x^3-3x^2y+3xy^2-y^3\)
  4. Keep factor powers balanced The powers of the two factors change in opposite directions and always add to n. \(\displaystyle \left(x+y\right)^4:\quad\left(4,0\right),\left(3,1\right),\left(2,2\right),\left(1,3\right),\left(0,4\right)\)

Worked example

Complete the coefficient and factor-power data for the expansion below

\[\left(3a-2b\right)^3\]
  1. \(\displaystyle P_3=\left(1,3,3,1\right)\)The power 3 selects Pascal row P3.
  2. \(\displaystyle 3^{3}=27,\quad3\times3^{2}\times\left(-2\right)=-54,\quad3\times3\times\left(-2\right)^{2}=36,\quad\left(-2\right)^{3}=-8\)Apply the scalar powers and the alternating signs to each row entry
  3. \(\displaystyle C=\left(27,-54,36,-8\right)\)These four values form the coefficient row after the scalar factors are included
  4. \(\displaystyle \left(3a-2b\right)^3=27a^3-54a^2b+36ab^2-8b^3\)The power of a falls while the power of b rises, and each pair sums to 3.

Common mistakes

Using the wrong Pascal row
✕\(\displaystyle \left(x+y\right)^4\longrightarrow P_3=\left(1,3,3,1\right)\)✓\(\displaystyle \left(x+y\right)^4\longrightarrow P_4=\left(1,4,6,4,1\right)\)
The row index is the total binomial power, so power 4 requires P4.
Not alternating subtraction signs
✕\(\displaystyle \left(x-y\right)^4=x^4+4x^3y+6x^2y^2+4xy^3+y^4\)✓\(\displaystyle \left(x-y\right)^4=x^4-4x^3y+6x^2y^2-4xy^3+y^4\)
Each term containing an odd power of the subtracted factor has a negative sign
Moving both factor powers together
✕\(\displaystyle \left(x+y\right)^4:\quad\left(4,0\right),\left(5,1\right),\left(6,2\right),\left(7,3\right),\left(8,4\right)\)✓\(\displaystyle \left(x+y\right)^4:\quad\left(4,0\right),\left(3,1\right),\left(2,2\right),\left(1,3\right),\left(0,4\right)\)
One factor power decreases as the other increases, so every pair must add to 4.

Difficulty levels

introThis tier identifies short Pascal rows and their interior coefficients
standardThis tier matches coefficients to powers in short binomial expansions
challengeThis tier handles longer rows, scalar factors and alternating signs
examThis tier combines coefficient, sign and factor-power structure in compact questions

Questions about this topic

Why does the row index equal the binomial power?+

Pascal row P_n contains the coefficients for a binomial raised to power n.

How can the factor powers be checked?+

The two powers should move in opposite directions and add to the total binomial power in every term

Which terms are negative in a subtraction expansion?+

Terms with an odd power of the second, subtracted factor are negative; terms with an even power are positive

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 5: Differentiation

In this chapter

Evaluate limits and test continuityGradient of a tangent from first principlesDifferentiate power functions and evaluate a derivativeExpand and factor polynomial expressionsPascal rows and binomial coefficient structure5A Limits, first principles and the power rule: Simplify and combine algebraic fractionsEvaluate a polynomial or recover a coefficient parameterSubstitute values into a formula5A Limits, first principles and the power rule: Solve a linear equationTest or construct a smooth piecewise joinRead and construct a gradient function from a function graphAnalyse a cubic graph or determine its ruleEvaluate and determine function rulesEquation of a tangent or normal lineRecover a polynomial rule from derivative and tangent conditionsFind the gradient of a line through two pointsFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a pair of simultaneous linear equationsDetermine parameters in a polynomial identity5C Tangents, normals and derivative conditions: Solve a linear equationDifferentiate natural exponential functions and solve simple derivative conditionsExponential growth and decay rate models5D Exponential derivatives and rate models: Simplify and combine algebraic fractions5D Exponential derivatives and rate models: Round to decimal places and significant figuresDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsGeometric optimisation with a trigonometric modelSinusoidal models and their rates of changeSolve a practical right-angled-triangle problemPerimeter and area of standard shapes5E Trigonometric differentiation and rates: Round to decimal places and significant figuresApproximate a cubic root using Newton's method