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All courses Math III · A-APR.5 5 of 55
Apply the Binomial Theorem for expanding (x+y)^n using Pascal's Triangle or combinatorial reasoning.

Expand a binomial power using Pascal's Triangle

Problem
Expand this expression using Pascal's Triangle: (x + y)⁴.
Unsolved x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴ Open full size
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Hint

Use the Pascal's Triangle row for power \(4\): \(1, 4, 6, 4, 1\).

The powers of \(x\) decrease from \(4\) to \(0\), while the powers of \(y\) increase from \(0\) to \(4\).

Solution walkthrough

01

Choose the Pascal's Triangle row

\[n~=~4~\Rightarrow~1,4,6,4,1\]

Because the binomial is raised to the fourth power, use the row of coefficients for exponent \(4\).

02

Assign the powers of x and y

\[x^4,\ x^3y,\ x^2y^2,\ \text{xy}^3,\ y^4\]

In each term, the exponent of \(x\) drops by \(1\) while the exponent of \(y\) rises by \(1\).

03

Attach the coefficients

\[1x^4+4x^3y+6x^2y^2+4\text{xy}^3+1y^4\]

Match each term pattern with the coefficient from Pascal's Triangle.

04

Write the expanded expression

\[(x+y)^4~=~x^4+4x^3y+6x^2y^2+4\text{xy}^3+y^4\]

This is the full expansion using Pascal's Triangle.

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Another way

  1. You can expand \((x+y)^2\) first and then square again, but Pascal's Triangle gives the coefficients faster.

!

Common mistake

A common mistake is to use the right coefficients but assign the wrong powers, such as writing \(x^4+4x^2y^2+6xy^3+...\). The exponents must decrease and increase one step at a time.

x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴
x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴