Hover or tap any entry to see how it's built from the two entries above it — Pascal's rule in action.
Hover any entry to see how it's built.
How it's read: the entry in row n, position k (both counted from 0) is the binomial coefficient C(n, k) — the number of ways to choose k items from n. Row sums double every row, the triangle is perfectly symmetric, and its diagonals hide the counting, triangular and Fibonacci numbers — explore all of it on the other tabs.
Every entry is the sum of the two above it — that single rule generates everything on this page.
C(n, k) for every position k in your row. The bell shape is no accident — as n grows, this converges on the normal distribution (de Moivre–Laplace).
Why the alternating sum is 0: setting x = 1, y = −1 in (x + y)ⁿ gives 0ⁿ — so adding and subtracting the coefficients in turn always cancels out (for every row after row 0). The count of odd coefficients is 2 raised to the number of 1-bits in n's binary form, a consequence of Lucas' theorem.
One row of the triangle is one complete binomial expansion — its entries are the coefficients.
Every C(n, k) across the row, with your chosen k in orange. The middle of a row always dominates — choosing about half the items can be done the most ways.
Coin-flip odds: the chance of exactly k heads in n fair flips is C(n, k) ÷ 2ⁿ — your answer divided by all possible outcome sequences. That's why one row of Pascal's triangle is the binomial distribution for n flips.
Every combinations problem is one entry of Pascal's triangle — row n, position k.
Entries NOT divisible by m are colored; the blanks are where m divides the entry evenly.
Why fractals appear: Lucas' theorem says C(n, k) mod a prime p depends only on the base-p digits of n and k — so the pattern for the first p rows stamps itself into ever-larger copies, forever. For p = 2, almost every entry is eventually even: the colored (odd) share shrinks toward zero as the triangle grows, tracing the Sierpinski triangle, a fractal of dimension log 3 / log 2 ≈ 1.585.
Color every entry by its remainder when divided by m — and watch fractals fall out of arithmetic.
Pick a property to light it up on the triangle — each one is a theorem you can see.