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pascal-triangle

Pascal's Triangle is a triangular array where each entry is the sum of the two directly above it: row 0 starts with 1, row 1 is 1, 1, row 2 is 1, 2, 1, and so on. Each entry equals the binomial coefficient C(n, k) — also known as "n choose k". The triangle has many number-theoretic properties: rows sum to powers of 2, diagonals form Fibonacci numbers, prime rows have all entries divisible by p. The ZTools Pascal Triangle Generator builds N rows (up to 30) in the browser, highlights selected patterns, and outputs as plain text or LaTeX.

Use cases​

Compute binomial coefficients​

"How many ways to pick 3 from 10?" → row 10, column 3 → 120. Calculator does it instantly.

Teaching binomial expansion​

(a + b)⁵ expands using row 5 of the triangle: a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵. Visual makes it click.

Spot Fibonacci in the triangle​

Sum the shallow diagonals — get 1, 1, 2, 3, 5, 8, 13, ... (Fibonacci sequence). Mind-bending classroom moment.

Probability calculations​

Coin flip outcomes for N flips: row N gives the number of ways to get 0, 1, 2, ..., N heads. Useful for binomial distribution intuition.

How it works​

  1. Pick row count — 1 to 30. Above 30, numbers exceed JavaScript safe integer range — toggle BigInt for higher rows.
  2. Generate — Each row computed iteratively from the previous. C(n, k) = C(n−1, k−1) + C(n−1, k).
  3. Display — Triangular layout with consistent column widths. Toggle to monospace or LaTeX output.
  4. Highlight patterns — Optional overlays: even/odd (Sierpinski-like fractal), multiples of 3, prime-row pattern.

Examples​

Input: First 6 rows

Output: 1\n1 1\n1 2 1\n1 3 3 1\n1 4 6 4 1\n1 5 10 10 5 1

Input: Row sums (rows 0-5)

Output: 1, 2, 4, 8, 16, 32 — powers of 2. Row n sums to 2^n.

Input: Shallow diagonal sums

Output: 1, 1, 2, 3, 5, 8, 13, 21 — Fibonacci sequence appears.

Frequently asked questions​

Why does the triangle relate to binomials?

C(n, k) counts subsets of size k from n elements. Row n column k of the triangle = C(n, k). Equivalent definitions.

How big can N be?

30 with regular numbers. Beyond, use BigInt mode — supports arbitrary row count, but display becomes unwieldy past row 50.

What's the Sierpinski connection?

Colour the triangle entries by parity (even/odd). The pattern of odds forms a Sierpinski triangle fractal — visible from row 16 or so.

Are there generalisations?

Yes — Pascal's pyramid (3D), Pascal's tetrahedron, multinomial coefficients. This tool covers the standard 2D triangle.

Privacy?

All in browser.

Tips​

  • For binomial-expansion teaching, plot the triangle alongside (a+b)^n — students see the coefficients literally come from row n.
  • For probability work, row N entries give the count of N-flip outcomes by head count — one step from the binomial PMF.
  • For Fibonacci connection, draw the shallow diagonals and sum them — vivid demo of how patterns emerge from simple rules.
  • For BigInt mode (rows >30), expect display to wrap on narrow screens — view in landscape or download as text.

Try it now​

The full pascal-triangle runs in your browser at https://ztools.zaions.com/pascal-triangle — no signup, no upload, no data leaves your device.

Open the tool ↗

Last updated: 2026-05-06 · Author: Ahsan Mahmood · Edit this page on GitHub