Pascal’s Triangle
Make Pascal’s triangle, the triangle of numbers in which each entry is the sum of the two above it. Choose how many rows to show: with the numbers on up to 26 rows, or as a pattern of up to 256 rows coloured by each number’s remainder when divided by m (2 to 16). Highlight a hockey stick or a shallow diagonal and see its sum, show each row’s sum (always a power of 2), or blank some numbers to make a worksheet. Every number is an exact whole number, and you can look up any single entry C(n, k) for n up to 10,000. Download a PDF, a 300 DPI PNG, an SVG or every number as CSV. Nothing is uploaded.
100% private — everything runs in your browser. Everything runs in your browser; nothing is uploaded.
Print at 100% (Actual size). Real size prints on one sheet when the drawing fits; the check bar in the footer lets you confirm the scale. The SVG holds the drawing at the printed size. The CSV holds every number drawn with its remainder mod m.
Pascal’s triangle
Look up one entry
Look
Page
Runs in your browser with our own code: nothing you type is uploaded or saved. Every number is an exact whole number (BigInt), each row is added up from the row above, and the colours, highlights and sums come from those same numbers. You may use what you make here, including commercially, with no credit needed.
How to make Pascal’s triangle
- Pick a preset, or set the number of rows.
- Turn the numbers on or off, and colour the cells by their remainder mod m if you like.
- Choose a highlight (hockey stick or shallow diagonal), row sums or worksheet blanks.
- Choose Fitted to one page or Real size and the paper.
- Download the PDF and print at 100% (Actual size), or the PNG, SVG or CSV.
What you can use it for
- Maths worksheets: fill in the missing numbers
- Binomial coefficients for n choose k
- Teaching the hockey stick and diagonal patterns
- Colour-pattern art from remainders mod 2, 3, 5 …
- Plotter and laser art from the SVG
- Checking big entries exactly
The 15 presets
| Preset | Rows | Colouring | Highlight | Cells |
|---|---|---|---|---|
| Rows 0–9 with numbers | 0–9 | none | – | 55 |
| Rows 0–15 with numbers | 0–15 | none | – | 136 |
| Row sums: powers of 2 | 0–11 | none | row sums | 78 |
| Hockey stick: 1 + 3 + 6 + 10 + 15 = 35 | 0–8 | none | hockey stick = 35 | 45 |
| Shallow diagonal: 1 + 6 + 10 + 4 = 21 | 0–9 | none | diagonal = 21 | 55 |
| Odd numbers shaded (mod 2) | 0–15 | mod 2 | – | 136 |
| Coloured mod 3, numbers | 0–11 | mod 3 | – | 78 |
| Mod 2 art, 64 rows | 0–63 | mod 2 | – | 2,080 |
| Mod 3 art, 81 rows | 0–80 | mod 3 | – | 3,321 |
| Mod 5 art, 125 rows | 0–124 | mod 5 | – | 7,875 |
| Mod 7 art, 98 rows (circles) | 0–97 | mod 7 | – | 4,851 |
| Mod 4 art, 256 rows | 0–255 | mod 4 | – | 32,896 |
| Worksheet: fill in the inner numbers | 0–9 | none | 36 blanks | 55 |
| Worksheet: about half blank | 0–11 | none | 30 blanks | 78 |
| Real size: 12 mm cells, rows 0–9 | 0–9 | none | – | 55 |
Counts and sums are worked out by the tool from the numbers it draws.
Rows 0 to 12 and their sums
| Row n | Numbers | Sum | Power of 2 |
|---|---|---|---|
| 0 | 1 | 1 | 2^0 |
| 1 | 1 1 | 2 | 2^1 |
| 2 | 1 2 1 | 4 | 2^2 |
| 3 | 1 3 3 1 | 8 | 2^3 |
| 4 | 1 4 6 4 1 | 16 | 2^4 |
| 5 | 1 5 10 10 5 1 | 32 | 2^5 |
| 6 | 1 6 15 20 15 6 1 | 64 | 2^6 |
| 7 | 1 7 21 35 35 21 7 1 | 128 | 2^7 |
| 8 | 1 8 28 56 70 56 28 8 1 | 256 | 2^8 |
| 9 | 1 9 36 84 126 126 84 36 9 1 | 512 | 2^9 |
| 10 | 1 10 45 120 210 252 210 120 45 10 1 | 1,024 | 2^10 |
| 11 | 1 11 55 165 330 462 462 330 165 55 11 1 | 2,048 | 2^11 |
| 12 | 1 12 66 220 495 792 924 792 495 220 66 12 1 | 4,096 | 2^12 |
Hockey sticks
| Diagonal | Sum | Equals |
|---|---|---|
| C(1, 1) + … + C(5, 1) | 15 | C(6, 2) = 15 |
| C(2, 2) + … + C(6, 2) | 35 | C(7, 3) = 35 |
| C(2, 2) + … + C(9, 2) | 120 | C(10, 3) = 120 |
| C(3, 3) + … + C(8, 3) | 126 | C(9, 4) = 126 |
| C(4, 4) + … + C(10, 4) | 462 | C(11, 5) = 462 |
| C(5, 5) + … + C(12, 5) | 1,716 | C(13, 6) = 1,716 |
Shallow diagonals give the Fibonacci numbers
| Diagonal from row d | Numbers | Sum | Fibonacci number |
|---|---|---|---|
| 0 | 1 | 1 | F(1) = 1 |
| 1 | 1 | 1 | F(2) = 1 |
| 2 | 1 + 1 | 2 | F(3) = 2 |
| 3 | 1 + 2 | 3 | F(4) = 3 |
| 4 | 1 + 3 + 1 | 5 | F(5) = 5 |
| 5 | 1 + 4 + 3 | 8 | F(6) = 8 |
| 6 | 1 + 5 + 6 + 1 | 13 | F(7) = 13 |
| 7 | 1 + 6 + 10 + 4 | 21 | F(8) = 21 |
| 8 | 1 + 7 + 15 + 10 + 1 | 34 | F(9) = 34 |
| 9 | 1 + 8 + 21 + 20 + 5 | 55 | F(10) = 55 |
| 10 | 1 + 9 + 28 + 35 + 15 + 1 | 89 | F(11) = 89 |
| 11 | 1 + 10 + 36 + 56 + 35 + 6 | 144 | F(12) = 144 |
| 12 | 1 + 11 + 45 + 84 + 70 + 21 + 1 | 233 | F(13) = 233 |
| 13 | 1 + 12 + 55 + 120 + 126 + 56 + 7 | 377 | F(14) = 377 |
| 14 | 1 + 13 + 66 + 165 + 210 + 126 + 28 + 1 | 610 | F(15) = 610 |
| 15 | 1 + 14 + 78 + 220 + 330 + 252 + 84 + 8 | 987 | F(16) = 987 |
Odd numbers in the first rows
| Rows | Numbers | Odd numbers | Rule | Multiples of 3 |
|---|---|---|---|---|
| 0–1 | 3 | 3 | 3^1 = 3 | 0 |
| 0–3 | 10 | 9 | 3^2 = 9 | 2 |
| 0–7 | 36 | 27 | 3^3 = 27 | 9 |
| 0–15 | 136 | 81 | 3^4 = 81 | 58 |
| 0–31 | 528 | 243 | 3^5 = 243 | 288 |
| 0–63 | 2,080 | 729 | 3^6 = 729 | 1,318 |
| 0–127 | 8,256 | 2,187 | 3^7 = 2,187 | 6,060 |
| 0–255 | 32,896 | 6,561 | 3^8 = 6,561 | 25,016 |
Counted from the exact values; up to 256 rows, the first 2^k rows hold exactly 3^k odd numbers.
Middle numbers of big rows
| Row | Middle entry | Value | Digits |
|---|---|---|---|
| 20 | C(20, 10) | 184,756 | 6 |
| 100 | C(100, 50) | 10089134…12497256 | 30 |
| 200 | C(200, 100) | 90548514…36841320 | 59 |
| 1,000 | C(1,000, 500) | 27028824…21216320 | 300 |
| 2,000 | C(2,000, 1,000) | 20481516…91149120 | 601 |
| 10,000 | C(10,000, 5,000) | 15917902…53649120 | 3,009 |
Exact, from the look-up box; long values show the first and last digits.
What prints
| Download | What is in it |
|---|---|
| The triangle fitted to one page or at real size, vector cells with the numbers, colours and highlights, with the sums and a check bar | |
| PNG 300 DPI | The same sheet as a picture |
| SVG | The triangle at the printed size: cells as rectangles or circles, numbers as text |
| CSV | Every number drawn with its row, position and remainder |
For a cutter that needs DXF, use SVG to DXF; for squared paper to copy the triangle by hand, use Graph Paper.
Which tool to use
| You want | Use |
|---|---|
| Pascal’s triangle with numbers, colours, sums and worksheets | Pascal’s Triangle (this page) |
| The triangle of triangles, the chaos game and the carpet | Sierpinski Triangle |
| Rows of cells from a rule | Rule 30 |
| The Fibonacci numbers themselves | Fibonacci Sequence |
Other tools: Graph Paper prints grids for working by hand, the Random Number Generator picks rows to quiz and the Fractal Generator draws other self-similar patterns.
More free tools
Maths art, patterns and printables?
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Frequently asked questions
What is Pascal’s triangle?
A triangle of numbers that starts with 1 at the top; each row begins and ends with 1, and every other number is the sum of the two numbers above it. Row n (counting the top row as row 0) holds the binomial coefficients C(n, 0) … C(n, n), the number of ways to choose k things from n.
Why is it named after Pascal?
It is named after the French mathematician Blaise Pascal, whose treatise on the arithmetical triangle was published in 1665, after his death; other mathematicians had studied it centuries before him, in India and elsewhere (Wikipedia, Pascal’s triangle, read 13 November 2026).
Are the numbers exact?
Yes. The tool adds whole numbers with JavaScript’s BigInt, which never rounds, so even the 76-digit middle of row 255 is exact. Our tests check every entry of rows 0 to 255 against n! / (k! (n − k)!). A single C(n, k) can be looked up for n up to 10,000; the middle of row 10,000 has more than 3,000 digits.
Why do the rows add up to powers of 2?
Each number is passed down to the two numbers below it, so every row adds up to twice the row above: 1, 2, 4, 8 … Row n sums to 2^n. The table below shows it for rows 0 to 12, and our tests check all 256 rows.
What is the hockey stick pattern?
Add the numbers going down a diagonal from a 1 at the edge, such as 1 + 3 + 6 + 10 + 15 in the diagonal that starts at C(2, 2): the sum is the number just below and to the side of the last one, here 35 = C(7, 3). In symbols, C(r, r) + C(r + 1, r) + … + C(n, r) = C(n + 1, r + 1). The tool outlines the stick in red and the answer in blue, and our tests check the rule for every stick in the first 60 rows.
Where are the Fibonacci numbers?
Add the numbers along a shallow diagonal: start at the 1 at the left of row d, then go one row up and one place right each time. The sums are 1, 1, 2, 3, 5, 8, 13 … the Fibonacci numbers: the diagonal starting in row d adds up to F(d + 1). Our tests check every one of the first 256.
What does colouring by mod m show?
Each cell is coloured by the remainder when its number is divided by m. With m = 2 the odd numbers form a pattern of triangles within triangles; other values of m give other patterns. In the first 2^k rows there are exactly 3^k odd numbers, as the table below shows for up to 256 rows. Remainder 0 is left white.
How many rows can it show?
Up to 26 rows (rows 0 to 25) with the numbers on, never printed smaller than 3 pt, and up to 256 rows as a colour pattern. Every preset draws in under 2 seconds in our tests.
Can I make a worksheet?
Yes. Choose Blank every inner cell to keep only the 1s along the edges, or Blank about half the inner cells (a pattern number picks which). Print it, then print again with All numbers shown for the answer key.
What is in the CSV?
Every number drawn: its row, its position k, the exact value and its remainder mod m.
Can it print at real size?
Yes, when the drawing fits on one sheet: choose Real size and set the size. Every printed cell, square and line is within 0.2 mm of its true place in our tests, on US Letter and A4, as long as you print at 100% (Actual size); the check bar in the footer lets you confirm the scale. Bigger drawings print fitted to the page.
Is it free?
Yes. There is no sign-up, no watermark and no limit.
Is anything uploaded?
No. Every number is worked out by JavaScript in your browser. Nothing you type is sent anywhere or saved when you close the page.