What is Pascal's Triangle?

Use this calculator to compute Pascal's Triangle values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Pascal's Triangle used?

Sequences, calculus, physics, engineering, and related disciplines.

What formulas does this use?

All standard pascal's triangle formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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ALGEBRASequencesMathematics Calculator

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Pascal's Triangle

Each entry C(n,k) = n!/(k!(n-k)!) counts ways to choose k from n. Row n sums to 2^n; shallow diagonals give Fibonacci numbers.

Concept Fundamentals

C(n,k) = n!/(k!(n-k)!)

Binomial coeff

2^n

Row sum

C(n,k) = C(n-1,k-1) + C(n-1,k)

Recursive

In diagonals

Fibonacci

Did our AI summary help? Let us know.

The pattern appeared in China (Yang Hui ~1000 AD) before Pascal published in 1653. Coloring odd numbers black produces the Sierpinski triangle fractal. The hockey stick pattern: diagonal sums equal the number where the diagonal ends.

Key quantities

C(n,k) = n!/(k!(n-k)!)

Binomial coeff

Key relation

2^n

Row sum

Key relation

C(n,k) = C(n-1,k-1) + C(n-1,k)

Recursive

Key relation

In diagonals

Fibonacci

Key relation

Ready to run the numbers?

Why: Pascal's triangle gives coefficients for (x+y)^n, counts combinations, and connects to probability.

How: Each entry is the sum of the two above. Row n gives coefficients of (x+y)^n.

The pattern appeared in China (Yang Hui ~1000 AD) before Pascal published in 1653.Coloring odd numbers black produces the Sierpinski triangle fractal.

Sources:Wolfram MathWorldOEIS A007318

Run the calculator when you are ready.

Gateway to CombinatoricsBinomial coefficients and beyond

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COMBINATORICS

Pascal's Triangle — Gateway to Combinatorics

Generate the triangle, compute C(n,k), explore row sums, hockey stick, and binomial theorem connections.

Binomial →Fibonacci →

🔺 Common Examples — Click to Load

Mode

Number of Rows

pascal_triangle.sh

CALCULATED

$ calc --rows=8

Rows

Row Values

1, 7, 21, 35, 35, 21...

Row Sum (2^n)

128

Number of Elements

Pascal's Triangle Calculator

8 rows generated

2^7 = 128

numbervibe.com/calculators/mathematics/sequences/pascal-triangle

Pascal's Triangle

121

1331

14641

15101051

1615201561

172135352171

Row Values (Bar)

Row Sums Growth (2^n)

Coefficient Distribution (Doughnut)

Step-by-Step Breakdown

CONSTRUCTION

Row 01 (by definition)

Row 11 1

Row 21 2(=1+1) 1

Row 31 3(=1+2) 3(=2+1) 1

Row 41 4(=1+3) 6(=3+3) 4(=3+1) 1

Row 51 5(=1+4) 10(=4+6) 10(=6+4) 5(=4+1) 1

Row 61 6(=1+5) 15(=5+10) 20(=10+10) 15(=10+5) 6(=5+1) 1

Row 71 7(=1+6) 21(=6+15) 35(=15+20) 35(=20+15) 21(=15+6) 7(=6+1) 1

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🏒

The hockey stick: sum along a diagonal equals the number where it ends.

11ⁿ

Concatenating row n gives 11^n for small n (e.g., 14641 = 11^4).

Key Takeaways

C(n,k) = n!/(k!(n-k)!) — each entry is the number of ways to choose k from n
Row sum = 2^n — the nth row sums to 2 raised to the power n
Fibonacci in diagonals — add numbers along shallow diagonals to get 1,1,2,3,5,8...
Symmetry — C(n,k) = C(n,n-k); each row is symmetric around its center
The nth row gives coefficients of $(x+y)^n$ in the binomial expansion

Did You Know?

📜Pascal didn't discover it — the pattern appeared centuries earlier in China (Yang Hui ~1000 AD), India, and Persia (Al-Karaji). Pascal published his treatise in 1653.Source: Wikipedia

🏒The 'hockey stick' pattern: start at any 1 on the edge, go diagonally, sum the numbers — the total appears in the next row.Source: Cut-the-Knot

11ⁿFor small n, concatenating row n gives 11^n. Row 4: 1,4,6,4,1 → 14641 = 11^4. Breaks down when digits carry.Source: Khan Academy

🔺Coloring odd numbers black and even white produces the Sierpinski triangle fractal pattern.Source: Fractals

📊C(n,k) connects to the binomial theorem: (x+y)^n expansion uses row n coefficients. Used in probability (binomial distribution) and combinatorics.Source: NCTM

🎲Probability distributions: P(k successes in n trials) uses C(n,k). Pascal's triangle gives these coefficients directly.Source: Statistics

How Pascal's Triangle Works

1. Binomial Coefficients

The number at (n,k) = $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ — ways to choose k items from n.

2. Recursive Construction

$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$ — each entry is the sum of the two above.

3. Binomial Theorem

$(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$ — row n gives expansion coefficients.

Expert Tips

Finding Patterns

Look for Fibonacci in diagonals, powers of 11 in concatenated rows, and Sierpinski when coloring odd/even.

Combinatorics Shortcuts

Use Pascal's triangle instead of computing factorials — C(n,k) is at row n, position k.

Connection to Probability

Binomial distribution: P(k successes in n trials) ∝ C(n,k). Row n gives all coefficients.

Binomial Expansion

For (x+y)^n, use row n. Row 6: 1,6,15,20,15,6,1 → (x+y)^6 = x^6 + 6x^5y + 15x^4y^2 + ...

Comparison Table

Feature	This Calculator	Manual	Programming
Visual triangle	✅	⚠️ Tedious	⚠️ Code needed
Binomial C(n,k)	✅	❌ Factorial overflow risk	✅
Row sum verification	✅	❌	✅
Step-by-step	✅	❌	❌
Charts	✅	❌	⚠️ Code needed
Copy & Share	✅	❌	❌
Educational content	✅	❌	❌

FAQ

What is C(n,k)?

C(n,k) = n!/(k!(n-k)!) is the number of ways to choose k items from n distinct items. Same as "n choose k" or combinations.

Why does each row sum to 2^n?

There are 2^n subsets of an n-element set. Each subset has some size k, and C(n,k) counts subsets of size k. Sum over k = 2^n.

What is the hockey stick pattern?

Start at a 1 on the left or right edge, move diagonally inward. The sum of numbers along that diagonal equals the number where the diagonal ends.

How does Pascal relate to probability?

Binomial distribution: P(k successes in n trials) uses C(n,k). Pascal's triangle gives these coefficients.

What is the Sierpinski connection?

Color odd numbers one color, even another. A fractal (Sierpinski triangle) emerges.

Why is it called Pascal's Triangle?

Blaise Pascal wrote a comprehensive treatise in 1653, though the pattern was known earlier in China, India, and Persia.

Can I use negative n or k?

For this calculator, n and k are non-negative integers. Generalized binomial coefficients exist for real n.

What is the maximum n?

We support n up to 100 for binomial coefficients. For full triangle, up to 30 rows for performance.

Where do Fibonacci numbers appear?

Add numbers along shallow diagonals (top-left to bottom-right). The sums 1,1,2,3,5,8... form the Fibonacci sequence.

Infographic Stats

~1000 AD

Discovered (China/India)

20+

Known patterns

Fibonacci

In diagonals

2^n

Row sum

Official Sources

Wolfram MathWorld ↗

Pascal's Triangle properties

Updated: 2026-02-01

OEIS A007318 ↗

Binomial coefficient sequence

Binomial theorem & series

Updated: 2026-01-15

Disclaimer: This calculator provides mathematically precise binomial coefficients. For very large n, floating-point limits may apply. Educational use.

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⬅️Jump in and explore the concept!

Pascal's Triangle

Pascal's Triangle — Gateway to Combinatorics

🔺 Common Examples — Click to Load

Mode

Pascal's Triangle

Row Values (Bar)

Row Sums Growth (2^n)

Coefficient Distribution (Doughnut)

Step-by-Step Breakdown

🧮 Fascinating Math Facts

Key Takeaways

Did You Know?

How Pascal's Triangle Works

1. Binomial Coefficients

2. Recursive Construction

3. Binomial Theorem

Expert Tips

Finding Patterns

Combinatorics Shortcuts

Connection to Probability

Binomial Expansion

Comparison Table

FAQ

What is C(n,k)?

Why does each row sum to 2^n?

What is the hockey stick pattern?

How does Pascal relate to probability?

What is the Sierpinski connection?

Why is it called Pascal's Triangle?

Can I use negative n or k?

What is the maximum n?

Where do Fibonacci numbers appear?

Infographic Stats

Official Sources

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