Pascal's Triangle Explained: How Every Number Adds Up to a Binomial Coefficient

Pascal's triangle is one of those objects that looks like a doodle and turns out to run half of combinatorics. You start with a single 1 at the top, keep adding numbers, and within a few rows you are staring at the coefficients of a binomial expansion, the count of every poker hand, and a hidden copy of the Fibonacci sequence. None of that is a coincidence. It all falls out of one tiny rule.

This post walks through that rule, builds a few rows by hand, reads a binomial coefficient straight off the grid, and points at the patterns that make the triangle worth keeping in your back pocket.

The one rule that builds the whole thing

Here is the entire construction: each entry is the sum of the two numbers directly above it. Put a 1 at the very top. Every row after that starts and ends with 1, and each inner number is filled in by adding its two upstairs neighbors.

Watch row 4 grow out of row 3, which is 1 3 3 1:

Left edge: 1
3 + 1 = 4
3 + 3 = 6
1 + 3 = 4
Right edge: 1

So row 4 is 1 4 6 4 1. That is the whole algorithm. There is no special case, no lookup table, no formula you have to memorize first. Add two numbers, write the result below and between them, repeat.

One thing trips people up immediately: the rows are counted from zero. The lonely 1 at the top is row 0. The line 1 1 is row 1. So 1 4 6 4 1 is row 4, even though it is the fifth line your eye counts down the page. Hold onto that, because the row number is about to mean something exact.

Each row is a list of binomial coefficients

Now the payoff. The entry in row n at position k — counting both from 0 — is exactly C(n, k), the binomial coefficient, the number of ways to choose k items from n. The triangle is a combinations table that builds itself.

Look at row 4 again: 1 4 6 4 1. Lined up against the coefficients, that reads

C(4,0)  C(4,1)  C(4,2)  C(4,3)  C(4,4)
  1       4       6       4       1

So if a probability problem hands you C(4, 2), you do not touch a factorial. You walk to row 4, count over to position 2, and read 6. Want C(5, 2)? Build row 5 — 1 5 10 10 5 1 — and the third number is 10. The triangle answers "how many ways can I choose k things from n" by position, not by arithmetic.

Why does the add-the-two-above rule produce combinations? Because choosing k items from n splits into two cases: either you take the last item (then you need k−1 more from the remaining n−1) or you skip it (then you need all k from the remaining n−1). That is exactly C(n−1, k−1) + C(n−1, k), the two cells sitting above. The rule and the meaning are the same statement wearing two costumes.

Reading a binomial expansion off the triangle

This connection is the reason the triangle shows up in algebra class. The coefficients in the expansion of (a + b) raised to the power n are precisely row n.

Take (a + b) to the 4th power. Instead of multiplying four brackets together and collecting terms, read row 4:

$$(a+b)^4 = 1a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 1b^4$$

The 1 4 6 4 1 out front is row 4, the powers of a step down, the powers of b step up. Need the cube? Row 3 is 1 3 3 1, so (a + b)³ = a³ + 3a²b + 3ab² + b³. No expansion grind, no sign errors — just copy the row.

A worked example: finding C(4, 2)

Let me build it from scratch so the whole chain is visible:

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 4, position 2 (start counting at 0: the first 1 is position 0, the 4 is position 1, the 6 is position 2). So C(4, 2) = 6. That matches the formula 4! / (2! · 2!) = 24 / 4 = 6, and it matches the middle coefficient in (a + b)⁴. Three different questions, one number, no calculator.

I keep a triangle pinned in my notes for exactly this reason. The first time I had to expand (a + b) to the 6th power under time pressure, I started multiplying brackets and lost the thread around the fourth term. Then a teacher drew six rows in the corner of the board, pointed at 1 6 15 20 15 6 1, and the whole expansion was just sitting there. I have never expanded a binomial the slow way since. If you want to generate the rows without drawing them, the Pascal's triangle generator builds as many as you ask for and lets you highlight one row to read off its coefficients.

The patterns hiding in plain sight

The triangle is generous with bonus structure once you know where to look.

Powers of 2 in the row sums. Add every number in a row and you always get a power of two. Row 0 sums to 1, row 1 to 2, row 2 to 4, row 3 to 8 — in general, row n sums to 2 to the power n. The reason is clean: a row counts every way to choose 0, 1, 2, … up to n items, and the total number of subsets of an n-element set is 2ⁿ. The row sum is just "all subsets, counted by size."

Fibonacci along the shallow diagonals. Tilt your head and add the numbers along the gentle diagonals — 1, then 1, then 1 + 1, then 1 + 2, then 1 + 3 + 1 — and you get 1, 1, 2, 3, 5, 8, the Fibonacci sequence. It hides in the triangle because both objects obey an add-the-previous-pieces rule, so the same numbers fall out. You can cross-check the run against a dedicated Fibonacci sequence generator if you want to see the diagonals line up.

Counting and triangular numbers along the straight diagonals. The first diagonal is all 1s. The second is 1, 2, 3, 4, … — the plain counting numbers. The third is 1, 3, 6, 10, 15 — the triangular numbers, each one a running total of the diagonal before it. Combinations, counting numbers, triangular numbers, and Fibonacci all share the same little grid.

Where the triangle earns its keep

You will reach for Pascal's triangle whenever you need a binomial expansion without the algebra, a combination count without a factorial, or a quick way to show students why C(n, k) and (a + b)ⁿ are secretly the same idea. The build rule is trivial to teach, the symmetry is visible at a glance, and the row-sum-equals-2ⁿ pattern makes a satisfying classroom reveal.

A couple of honest gotchas. Count rows and positions from 0, or your C(n, k) lookups will be off by one. And remember the numbers grow fast — the coefficients blow past what ordinary floating-point arithmetic keeps exact within a few dozen rows, so a value like C(100, 50) needs big-integer math to stay correct to the last digit. A triangle drawn by hand is fine for homework; deep rows want a tool that does exact arithmetic.

For the build-row-by-row visual, exact deep coefficients, and one-click reading of any row as a binomial expansion, the Pascal's triangle generator handles all of it in the browser. From there the same numbers connect outward — into combinations, the binomial theorem, and the number patterns that keep turning up once you start adding two cells at a time.

Made by Toolora · Updated 2026-06-13