Pascal\'s Triangle Generator

What Is Pascal's Triangle?

Pascal's triangle is a triangular array of numbers where each entry is the sum of the two entries directly above it. Row $n$, position $k$ holds the binomial coefficient $\binom{n}{k}$. The triangle starts with row 0 at the top (a single 1), and each subsequent row has one more entry than the previous.

        1
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1
   1 5 10 10 5 1

In this calculator, the interactive Pascal's triangle renders up to row 30. Click any cell to set $n$ and $r$ and compute the corresponding binomial coefficient.

Key Patterns in Pascal's Triangle

Row Sums

The sum of all entries in row $n$ equals $2^n$—the total number of subsets of an $n$-element set:

Symmetry

Each row is a palindrome. This reflects the identity:

Choosing $k$ items to include is the same as choosing $n-k$ items to exclude.

Diagonals

Hockey Stick Pattern

If you take any diagonal of consecutive entries and sum them, the result equals the entry one row down and one position to the side:

This is called the "hockey stick" identity because the pattern of summed entries plus the result forms a hockey stick shape in the triangle.

Connection to the Binomial Theorem

Row $n$ of Pascal's triangle gives exactly the coefficients of the expansion of $(a + b)^n$:

For example, row 4 is [1, 4, 6, 4, 1], giving:

This makes Pascal's triangle an indispensable tool for expanding binomial expressions without tedious multiplication.

Applications

• Probability: The binomial distribution uses row entries directly. The probability of getting exactly $k$ heads in $n$ fair coin flips is $\frac{\binom{n}{k}}{2^n}$.
• Combinatorics: Every entry $\binom{n}{k}$ answers the question "how many ways can I choose $k$ items from $n$?"
• Number theory: The triangle encodes divisibility patterns. Row $p$ (where $p$ is prime) has all interior entries divisible by $p$.
• Fractal geometry: Coloring odd entries in Pascal's triangle produces the Sierpiński triangle, a self-similar fractal.

Frequently Asked Questions