Pascal's Triangle Generator – Build the Triangle

Generate Pascal's triangle up to any number of rows with our free online tool. Explore binomial coefficients, combinations, and mathematical patterns.

Number of Rows

Examples:

Understanding Pascal's Triangle

Pascal's triangle is one of the most elegant patterns in mathematics. Start with a single 1 at the top. Each row below adds one more number. Every number is the sum of the two numbers directly above it. The edges are always 1. What emerges is a triangular array packed with mathematical relationships.

Named after French mathematician Blaise Pascal, who studied it extensively in 1653, the triangle was actually known centuries earlier in China, Persia, and India. Chinese mathematician Yang Hui described it in 1261, and Persian polymath Omar Khayyam worked with it in the 11th century.

How Pascal's Triangle Works

Building the Triangle

Row 0 starts with just [1]. Row 1 is [1, 1]. From there, each interior number equals the sum of the two numbers above it. Think of it as adding neighbors from the previous row.

Row 0: 1

Row 1: 1 1

Row 2: 1 2 1

Row 3: 1 3 3 1

Row 4: 1 4 6 4 1

Row 5: 1 5 10 10 5 1

The Formula Behind It

Each entry in Pascal's triangle is a binomial coefficient. The number at position k in row n equals "n choose k" – the number of ways to pick k items from n items.

C(n,k) = n! / (k! × (n-k)!)

Row 5, position 2: C(5,2) = 5!/(2!×3!) = 120/(2×6) = 10

Patterns Hidden in the Triangle

Row Sums Are Powers of 2

Add up all numbers in row n, and you get 2^n. Row 4 sums to 1+4+6+4+1 = 16 = 2^4.

Diagonals Reveal Counting Numbers

The first diagonal is all 1s. The second diagonal counts 1, 2, 3, 4, 5... The third diagonal gives triangular numbers: 1, 3, 6, 10, 15...

Fibonacci Sequence

Add numbers along shallow diagonals and you get the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13...

Symmetry

Every row reads the same forwards and backwards. The triangle is perfectly symmetric down its center line.

Worked Examples

Example 1: Generate 6 Rows

Build Pascal's triangle with 6 rows (row 0 through row 5)

Row 0: 1

Row 1: 1 1

Row 2: 1 2 1

Row 3: 1 3 3 1

Row 4: 1 4 6 4 1

Row 5: 1 5 10 10 5 1

Total numbers: 21 | Largest: 10

Example 2: Find C(7,3)

Use Pascal's triangle to find the binomial coefficient "7 choose 3"

Look at row 7, position 3 (0-indexed)

Row 7: 1 7 21 35 35 21 7 1

Position 3 = 35

C(7,3) = 7!/(3!×4!) = 5040/(6×24) = 35

Example 3: Expand (x+y)^4

Use row 4 of Pascal's triangle as coefficients

Row 4 coefficients: 1 4 6 4 1

(x+y)^4 = 1x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 1y^4

= x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4

Example 4: Sum of Row 8

Find the sum of all numbers in row 8 without adding them

Sum of row n = 2^n

Sum of row 8 = 2^8 = 256

Verify: 1+8+28+56+70+56+28+8+1 = 256

Quick Fact

Pascal's triangle contains the tetrahedral numbers in its fourth diagonal: 1, 4, 10, 20, 35... These count the number of spheres needed to build a tetrahedron (triangular pyramid). The 10 in row 5 represents a pyramid of 10 cannonballs – 1 on top, then 3, then 6 at the base.

Frequently Asked Questions

What is Pascal's triangle used for?

Pascal's triangle gives binomial coefficients for expanding (x+y)^n. It's used in probability to find combinations, in algebra for polynomial expansion, and in combinatorics for counting problems. Statisticians use it for binomial distributions.

How do I find a specific number in Pascal's triangle?

Use the formula C(n,k) = n!/(k!(n-k)!), where n is the row number and k is the position (both starting from 0). For row 6, position 2: C(6,2) = 720/(2×24) = 15.

Why are the edges always 1?

The edges represent C(n,0) and C(n,n) – choosing nothing or choosing everything. There's exactly one way to do either, so these values are always 1. Mathematically, n!/(0!×n!) = 1.

Can Pascal's triangle have negative numbers?

Not in the standard triangle. Every entry is a positive counting number. However, extended versions exist for negative row indices, but those are advanced mathematical constructs.

What's the largest number I can generate?

This calculator handles up to 100 rows. The middle number in row 100 is C(100,50), which has about 30 digits. Beyond that, numbers grow astronomically – row 1000's center has nearly 300 digits.

Is there a pattern for odd and even numbers?

Yes. Color the odd numbers black and even numbers white, and you get the Sierpinski triangle fractal. This self-similar pattern emerges from the simple addition rule.

Other Free Tools

Search tools