The rule
each entry is the sum of the two above it
Why it works
Pascal's triangle starts with a single 1. Every row begins and ends with 1, and every other entry is the sum of the two directly above. Row n gives the coefficients of (a + b)ⁿ.
How to do it by hand
- Write a single 1 for the first row
- Start and end each new row with 1
- Every other entry is the sum of the two above it
- Row n contains the combinations of n items
What is worth knowing
The triangle is a filing cabinet of patterns. Each row adds to a power of two, because choosing any subset of n items gives 2ⁿ possibilities. The diagonals hold the counting numbers, then the triangular numbers, then the tetrahedral ones. Shading the odd entries produces the Sierpiński triangle, a fractal that nobody put there. And the whole thing predates Pascal by centuries: it appears in Chinese, Indian and Persian texts, which is why it is also called Yang Hui's triangle or Khayyam's triangle depending on where you are.