- Expand the rows of Pascal’s triangle
Pascal’s Triangle
Pascal’s triangle is a triangular arrangement of numbers where each row contains the coefficients for expanding binomials like [latex](a + b)^n[/latex]. Each number in the triangle is the sum of the two numbers directly above it. Understanding Pascal’s triangle helps you quickly find coefficients for binomial expansions.
What is Pascal’s Triangle?
Pascal’s triangle starts with 1 at the top and builds downward. Each row corresponds to the power of a binomial expansion.
[latex]\begin{array}{cccccccc} & & & & 1 & & & & \\[6pt] & & & 1 & & 1 & & & \\[6pt] & & 1 & & 2 & & 1 & & \\[6pt] & 1 & & 3 & & 3 & & 1 & \\[6pt] \cdots & & \cdots & & \cdots & & \cdots & \end{array}[/latex]
The triangle is named after French mathematician Blaise Pascal, but it was known centuries earlier in China and other cultures.
To find any entry in Pascal’s triangle:
- Start with 1’s on both edges of every row
- For interior numbers, add the two numbers directly above
- Each row has one more entry than its row number
- Row [latex]n[/latex] has [latex]n + 1[/latex] entries