Pascal's Triangle — Pattern, Properties & Uses

Need the coefficients of $(a+b)^n$ without multiplying the whole thing out? Pascal's triangle generates them with one local rule: each interior entry is the sum of the two entries directly above it, and the edges are all $1$. Counting the top as row $0$, row $n$ holds the coefficients of $(a+b)^n$.

When to use this method

Reach for the triangle when you want binomial coefficients, a quick binomial expansion, simple combination counts, or a check on an expansion you got another way. It is fastest for small to moderate $n$, where building rows by hand beats computing each $\binom{n}{k}$ separately. The first few rows:

The steps

1. Start with 1. Place a single $1$ at the top.
2. Keep 1s on the edges. Every new row begins and ends with $1$.
3. Add the two numbers above. Each interior entry is the sum of its two upper neighbors.
4. Read the pattern. With the top as row $0$, row $n$ gives the coefficients for $(a+b)^n$.

For example, in row $4$ the middle $6$ comes from $3 + 3$, and the $4$ beside it from $1 + 3$ — so each row is built straight from the row before it, no separate formula needed.

Why a row equals the binomial coefficients

Row $n$ can also be written with combinations:

where $\binom{n}{k}$ counts the ways to choose $k$ objects from $n$. That is why row $4$,

gives

and why the triangle ties algebra to counting.

The whole procedure on one example

Expand $(x+y)^5$. With the top as row $0$, row $5$ is

Match those coefficients with descending powers of $x$ and ascending powers of $y$:

The triangle supplies the coefficients, but you still place the powers in order: the exponent of $x$ falls from $5$ to $0$ while the exponent of $y$ rises from $0$ to $5$.

Where each step tends to stall, and how to check

Step 3 (adding above): only the two entries directly above a position count, never numbers off to the side. Step 4 (row numbering): if a source starts rows at $1$ instead of $0$, the coefficient row for $(a+b)^n$ is labeled differently, so fix your convention first. Placing powers: the triangle alone is not the full expansion — write the powers correctly too.

Two properties give you fast self-checks. Each row is symmetric, since $\binom{n}{k} = \binom{n}{n-k}$, so row $5$ reads $1,5,10,10,5,1$ the same both directions. And the row sum is $2^n$:

If your row $5$ does not add to $32$, something went wrong.

Build row $6$ from row $5$, then use it to expand $(m+n)^6$ — a clean way to drill both halves of the skill: generating the coefficients and placing the powers. In school math, Pascal's triangle usually appears right before or alongside the binomial theorem, and it stays useful as a quick coefficient check long after.