Pascal's Triangle Calculator

Number of rows N	8
Column k	3

Number of rows N

Column k

Last updated: 2026-07-02

Definition

Pascal's triangle is a triangular array in which every entry is a binomial coefficient. The apex is a single $1$ (row $0$ ), and each row after it begins and ends with $1$ . Row $n$ lists the coefficients:

\binom{n}{0}, \binom{n}{1}, \ldots, \binom{n}{n}

These are precisely the numbers that appear when $(a + b)^n$ is expanded, which is why the triangle sits at the meeting point of combinatorics and algebra.

Construction

Each interior entry is the sum of the two entries directly above it. This is Pascal's rule:

\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}

Because the two outer diagonals are always $1$ (there is exactly one way to choose none or all of the items, so $\binom{n}{0} = \binom{n}{n} = 1$ ), the whole triangle can be built by addition alone, starting from the apex:

Worked example

Build $6$ rows and read the entry at column $k = 2$ in the bottom row. With $N = 6$ , the bottom row has row index $N - 1 = 5$ :

1,\ 5,\ 10,\ 10,\ 5,\ 1

The entry at $k = 2$ is the binomial coefficient $\binom{5}{2}$ :

\binom{5}{2} = \frac{5!}{2!\,(5-2)!} = \frac{120}{2 \cdot 6} = 10

The same value follows from Pascal's rule using the row above ( $1, 4, 6, 4, 1$ ):

\binom{5}{2} = \binom{4}{1} + \binom{4}{2} = 4 + 6 = 10

Row sums

The entries in row $n$ always add up to a power of two:

\sum_{k=0}^{n} \binom{n}{k} = 2^n

This comes straight from the binomial theorem with $a = b = 1$ : $(1 + 1)^n = \sum_k \binom{n}{k} = 2^n$ . For the row above, $1 + 5 + 10 + 10 + 5 + 1 = 32 = 2^5$ , a quick way to confirm a row was copied correctly.

Patterns

Several well-known sequences are hidden in the triangle. The outermost diagonal is all $1$ s; the next diagonal counts the natural numbers $1, 2, 3, 4, \ldots$ ; the third gives the triangular numbers $1, 3, 6, 10, \ldots$ . Summing along the shallow diagonals produces the Fibonacci sequence. Every row is symmetric because

\binom{n}{k} = \binom{n}{n-k}

so it reads the same forwards and backwards — choosing which $k$ items to include is the same as choosing which $n - k$ to leave out.

Frequently Asked Questions (FAQ)

What is Pascal's triangle?

Pascal's triangle is a triangular array of numbers where each entry is a binomial coefficient. The top is a single 1 (row 0), and each following row starts and ends with 1. Row n reads C(n, 0), C(n, 1), …, C(n, n) — the coefficients that appear when (a + b)ⁿ is expanded. For example, row 4 is 1, 4, 6, 4, 1, matching (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.

How is each entry built?

Every interior entry is the sum of the two entries directly above it. This is Pascal's rule: C(n, k) = C(n − 1, k − 1) + C(n − 1, k). Starting from the 1 at the apex, you can build the whole triangle by addition alone, without ever computing a factorial. The edges are always 1 because C(n, 0) = C(n, n) = 1.

Why does each row add up to a power of 2?

The entries in row n sum to 2ⁿ. This follows from setting a = b = 1 in the binomial theorem: (1 + 1)ⁿ = Σ C(n, k) = 2ⁿ. So row 0 sums to 1, row 1 to 2, row 2 to 4, and row 10 to 1024. It is a quick way to check that a row has been written correctly.

What patterns hide in Pascal's triangle?

Several. The outermost diagonal is all 1s; the next diagonal counts 1, 2, 3, 4, … (the natural numbers); the third diagonal gives the triangular numbers 1, 3, 6, 10, …. Summing along the shallow diagonals produces the Fibonacci sequence. Each row is also symmetric because C(n, k) = C(n, n − k), so it reads the same left to right and right to left.

Pascal's Triangle Calculator

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Pascal's Triangle Calculator