Pascal's Triangle — Pattern, Properties & Uses

Pascal's triangle is the triangular number pattern where each interior entry is the sum of the two entries above it. If you count the top as row $0$, row $n$ gives the coefficients of $(a+b)^n$, which is why this pattern shows up so often in algebra and counting.

If you label the top as row $0$, the first few rows are:

Row $0$: $1$

Row $1$: $1,\ 1$

Row $2$: $1,\ 2,\ 1$

Row $3$: $1,\ 3,\ 3,\ 1$

Row $4$: $1,\ 4,\ 6,\ 4,\ 1$

That one rule creates a pattern that is easy to build by hand and useful far beyond the diagram itself.

How to build Pascal's triangle

The rule is local: each interior entry depends only on the two entries above it. For example, in row $4$, the middle $6$ comes from

$$3 + 3 = 6$$

and the $4$ next to it comes from

$$1 + 3 = 4$$

So you can generate each new row from the row before it, without memorizing a separate formula.

Why Pascal's triangle matches binomial coefficients

Pascal's triangle is not just a visual pattern. If the top is row $0$, then row $n$ gives the coefficients of $(a+b)^n$.

The same row can also be written with combinations:

$$\binom{n}{0},\ \binom{n}{1},\ \binom{n}{2},\ \dots,\ \binom{n}{n}$$

Here, $\binom{n}{k}$ means "the number of ways to choose $k$ objects from $n$ objects." That is why the triangle connects algebra and counting.

For example, row $4$ is

$$1,\ 4,\ 6,\ 4,\ 1$$

so

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

This is the main reason many students meet Pascal's triangle when they study the binomial theorem.

Worked example: expand $(x+y)^5$

Use Pascal's triangle to expand

$$(x+y)^5$$

If the top is row $0$, then row $5$ is

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

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

$$(x+y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5$$

This example shows the key idea: Pascal's triangle gives the coefficients, but you still need to place the powers in order. The exponent of $x$ starts at $5$ and decreases to $0$, while the exponent of $y$ starts at $0$ and increases to $5$.

Properties worth remembering

One important property is symmetry. In row $5$, the numbers read the same from left to right:

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

Another useful property is the row sum. If the top is row $0$, then the entries in row $n$ add up to $2^n$. For row $5$,

$$1+5+10+10+5+1 = 32 = 2^5$$

That pattern is helpful for quick checks. If your row $5$ does not add to $32$, something went wrong.

Common Pascal's triangle mistakes

One common mistake is mixing up the row number. If one source starts counting rows at $1$ instead of $0$, the coefficient row for $(a+b)^n$ will be labeled differently.

Another mistake is assuming the triangle gives the full expansion by itself. It gives the coefficients, but you still need to write the powers correctly.

A third mistake is adding entries that are not directly above the target position. Each interior number comes from exactly two neighbors in the row above.

When Pascal's triangle is used

Pascal's triangle is used to expand binomials, read binomial coefficients, count combinations, and recognize simple probability patterns. In school math, it often appears before or alongside the binomial theorem.

It is also useful as a quick check. If you already expanded $(a+b)^n$ another way, the coefficients should match the corresponding row of the triangle.

Try a similar problem

Build row $6$ from row $5$, then use it to expand $(m+n)^6$. That is a clean way to practice both parts of the idea: generating the coefficients and placing the powers correctly.