Pascal’s Triangle Calculator

Pascal’s triangle starts with 1 at the top, and every other number is the sum of the two numbers above it. Row n lists the binomial coefficients C(n, 0) to C(n, n): row 4 is 1, 4, 6, 4, 1, so (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴.

How to use the Pascal's triangle calculator

1. Type a row number n (the single 1 at the top is row 0).
2. Add a position k to get one entry, C(n, k), or leave it empty for the whole row.
3. Switch to Binomial expansion to expand (x + y)ⁿ, or to Odd numbers to see the Sierpiński triangle hidden in the pattern.

How the triangle is built

Each row starts and ends with 1. Every inside number is the sum of the two numbers just above it. This is Pascal’s rule: \(\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\).

Patterns to look for

• Row sums double each time: row n adds to 2ⁿ.
• Diagonals: the second diagonal is 1, 2, 3, 4, … (counting numbers); the third is 1, 3, 6, 10, … (triangular numbers).
• Powers of 11: rows 0 to 4 read as 1, 11, 121, 1331, 14641.
• Symmetry: each row reads the same forwards and backwards, because C(n, k) = C(n, n − k).
• Odd entries form the Sierpiński triangle, a fractal.

Pascal's triangle and combinations

Entry k of row n counts the ways to choose k things from n: \(\binom{n}{k} = \frac{n!}{k!\,(n-k)!}\). For example, there are C(5, 2) = 10 ways to pick 2 toppings from 5. The same numbers are the coefficients in the binomial theorem, and they give coin-flip probabilities: in 4 flips, the chance of exactly 2 heads is 6/16.

Pascal's triangle practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you find a row of Pascal's triangle without drawing it?

Start with 1 and multiply by (n − k + 1)/k for each next entry. For row 5: 1, 1·5/1 = 5, 5·4/2 = 10, 10·3/3 = 10, 10·2/4 = 5, 5·1/5 = 1.

Why is the top row called row 0?

Because row n gives the coefficients of (x + y)ⁿ, and (x + y)⁰ = 1. Counting from 0 makes entry k equal C(n, k).

What is Pascal's triangle used for?

Expanding binomials, counting combinations, finding probabilities for repeated yes/no trials, and spotting number patterns such as triangular numbers and the Fibonacci numbers along shallow diagonals.

How large a row can the calculator handle?

Up to row 1,000. It uses exact whole-number arithmetic, so even entries with hundreds of digits are exact.

Who discovered Pascal's triangle?

The pattern was known centuries before Blaise Pascal, for example to Yang Hui in China and to mathematicians in Persia and India. Pascal wrote a famous treatise on it in the 1650s.

Sources

• OpenStax College Algebra 2e: Binomial Theorem
• Wolfram MathWorld: Pascal's Triangle

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