Pascal's triangle

Writing the coefficients of each power in a row, one row under another, gives a triangle: 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1; and so on.

Each row starts and ends with 1, and every inside number is the sum of the two numbers just above it. From 1 4 6 4 1 the next row is 1 5 10 10 5 1, and after it 1 6 15 20 15 6 1.

This array is called Pascal's triangle, after the French mathematician Blaise Pascal (1623-1662). In India it was known much earlier as Meru Prastara, given by Pingla.

Using the row 1 5 10 10 5 1 with the power patterns: (2x + 3y)⁵ = (2x)⁵ + 5(2x)⁴(3y) + 10(2x)³(3y)² + 10(2x)²(3y)³ + 5(2x)(3y)⁴ + (3y)⁵ = 32x⁵ + 240x⁴y + 720x³y² + 1080x²y³ + 810xy⁴ + 243y⁵.

Keep the whole of each term inside brackets: (2x)⁴ is 16x⁴, not 2x⁴. Forgetting the bracket is the most common slip in expansions.

The weakness: for (2x + 3y)¹² every row up to row 12 must be written first, and larger powers are worse. A rule that gives any row directly is needed.