Pascal's Triangle Generator

How to use this calculator

Type how many rows you want (row 0 through row n − 1) and the calculator builds the triangle by the same additive rule Pascal used: start each row with 1, end it with 1, and fill every spot in between by adding the two numbers sitting directly above it. The answer band shows the last row you asked for, along with its total.

The build rule

Row 0 is just 1. Every later row keeps 1 at both ends, and each interior entry is the sum of the two entries above it:

Six rows built this way look like:

Row 0:            1
Row 1:          1   1
Row 2:        1   2   1
Row 3:      1   3   3   1
Row 4:    1   4   6   4   1
Row 5:  1   5  10  10   5   1

Check row 4's middle entry: $6 = 3 + 3$, the sum of the two 3s directly above it in row 3.

Row n holds the binomial coefficients

Position $k$ of row $n$ (both counted from 0) equals $\binom{n}{k} = C(n, k)$ — the number of ways to choose $k$ items from a set of $n$. Row 4 is $\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3}, \binom{4}{4}$, which works out to 1, 4, 6, 4, 1. That is also why every row is symmetric: $\binom{n}{k} = \binom{n}{n-k}$.

Row sums double every time

Add up any row and you get a power of two: row 4 sums to $1 + 4 + 6 + 4 + 1 = 16 = 2^4$. Each entry of a row feeds into forming two entries of the next row (one to the left, one to the right), so the row total exactly doubles at every step, starting from row 0's total of 1.

Binomial expansion

The same numbers are the coefficients when you expand $(a + b)^n$. Row 2 gives 1, 2, 1, matching $(a+b)^2 = a^2 + 2ab + b^2$. Row 3 gives 1, 3, 3, 1, matching $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ — the reason this triangle shows up constantly in algebra and probability.

Hidden diagonals

Reading the triangle along its diagonals uncovers other famous sequences:

That last one is the neatest: summing entries along a shallow diagonal — one step left, one step up, repeated — reproduces the Fibonacci sequence, tying this triangle directly to the golden-ratio family of calculators.

Worked example — 6 rows

1. Row 0: 1.
2. Row 1: edges are 1, so 1 1.
3. Row 2: edges 1, interior $1 + 1 = 2$ → 1 2 1.
4. Row 3: edges 1, interior $1+2=3$ and $2+1=3$ → 1 3 3 1.
5. Row 4: edges 1, interior $1+3=4$, $3+3=6$, $3+1=4$ → 1 4 6 4 1.
6. Row 5: edges 1, interior $1+4=5$, $4+6=10$, $6+4=10$, $4+1=5$ → 1 5 10 10 5 1, summing to $32 = 2^5$.