🔺 Pascal Triangle Generator

What is the Pascal Triangle Generator?

Pascal's triangle is a triangular array of numbers named after the French mathematician Blaise Pascal, though it was studied centuries earlier in India, Persia, China and elsewhere. The construction rule is simple: write 1 at the apex (this is row 0), then build each new row so that it starts and ends with 1 and every entry in between equals the sum of the two entries immediately above it to the left and right. So row 1 is 1 1, row 2 is 1 2 1, row 3 is 1 3 3 1, row 4 is 1 4 6 4 1, and the pattern continues forever.

The deep reason the triangle matters is that its entries are exactly the binomial coefficients. The number in row n at position k (counting both from 0) equals C(n, k), read as n choose k, which is n! ÷ (k! × (n − k)!). This is the count of distinct ways to choose k items from a set of n, and it is also the coefficient of the term containing one variable raised to the k power when you expand (a + b) to the n power. For example, (a + b) squared = 1a² + 2ab + 1b², and those coefficients 1, 2, 1 are precisely row 2 of the triangle.

Because the additive rule and the choose formula describe the same numbers, the triangle ties together combinatorics, algebra and probability in one picture. The rows sum to powers of 2 (each row n adds up to 2 to the n), the second diagonal lists the counting numbers 1, 2, 3, 4 and the third lists the triangular numbers 1, 3, 6, 10. The triangle is symmetric left to right because choosing k items to keep is the same as choosing n − k items to leave behind, so C(n, k) = C(n, n − k).

When to use it

• Looking up a binomial coefficient C(n, k) quickly without typing out factorials by hand.
• Reading off the coefficients to expand a binomial like (a + b) raised to a power in algebra homework.
• Teaching or learning the additive build rule and the symmetry of the triangle in a classroom.
• Finding combinatorics values (number of ways to choose k from n) for probability problems.

How to use the Pascal Triangle Generator

1. Enter the number of rows you want, from 1 up to 25.
2. Read the centered triangle: row 0 is the single 1 at the top.
3. Click any number to see it written as a binomial coefficient C(n, k) with its meaning.
4. Use Copy text or Copy CSV to export the whole triangle for a spreadsheet or document.

Formula & method

Worked examples

The first eight rows of Pascal’s triangle and what each row tells you

Patterns hidden in the diagonals of the triangle

Common mistakes to avoid

• Counting rows starting at 1 instead of 0. The apex is row 0, not row 1. The same applies to positions within a row, which also start at 0. Counting from 1 throws off the C(n, k) lookup, since the single 1 at the top is C(0, 0).
• Forgetting the edges are always 1. Every row must begin and end with 1 because C(n, 0) and C(n, n) both equal 1. If your row does not start and end with 1, an entry has been added incorrectly.
• Confusing the additive rule direction. An inner entry is the sum of the two numbers above it, one to the left and one to the right, not the two beside it in the same row. Reading sideways gives the wrong value.
• Mixing up C(n, k) with the factorial n!. A triangle entry is a binomial coefficient C(n, k) = n! ÷ (k! × (n − k)!), not a plain factorial. The factorials cancel a lot, so the result is much smaller than n! alone.

Glossary

Pascal's triangle

A triangular array where each row starts and ends with 1 and every inner number is the sum of the two numbers above it.

Binomial coefficient

The value C(n, k), the number of ways to choose k items from n, equal to n! ÷ (k! × (n − k)!). It fills the triangle.

Row (n)

A horizontal line of the triangle. The top single 1 is row 0, so row n contains n + 1 entries.

Binomial expansion

Writing out (a + b) raised to a power. The coefficients of the terms are exactly the entries of row n of the triangle.

Factorial (n!)

The product of all whole numbers from 1 to n, with 0! defined as 1. It appears in the C(n, k) formula.

Frequently asked questions