A table built from nothing but addition hides both a fractal and a bell curve inside itself. Each cell is the sum of the two above it, and that same cell counts the ways of choosing k objects from n, the paths descending to it from the apex, and the coefficients in the expansion of (a+b)ⁿ. A row sums to 2ⁿ because that is the number of subsets of an n-element set. Shade only the odd entries and the Sierpinski triangle emerges, which Lucas’ theorem explains neatly: C(n,k) is odd exactly when every 1-bit of k sits where n also has a 1-bit.
A table built from nothing but addition hides both a fractal and a bell curve inside itself. Each cell is the sum of the two above it, and that same cell counts the ways of choosing k objects from n, the paths descending to it from the apex, and the coefficients in the expansion of (a+b)ⁿ. A row sums to 2ⁿ because that is the number of subsets of an n-element set. Shade only the odd entries and the Sierpinski triangle emerges, which Lucas’ theorem explains neatly: C(n,k) is odd exactly when every 1-bit of k sits where n also has a 1-bit.
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