What Is Pascal's Triangle and How Is It Related to Binomial Expansion?

Pascal's triangle is a triangular arrangement of numbers that provides the binomial coefficients used when expanding $(a+b)^n$. Row $n$ gives the coefficients of the expansion, counting the top row as row $0$.

This is IB Mathematics: Analysis and Approaches, SL 1.9: The binomial theorem. The method is especially suitable for an exact, non-calculator Paper 1 question, although it may also appear on Paper 2.

How Pascal's Triangle Is Constructed

Each row begins and ends with $1$. Every other number is found by adding the two numbers immediately above it:

For example, the middle $6$ in row $4$ comes from $3+3$ in row $3$.

This construction is recursive: once one row is known, the next can be produced without expanding any algebraic expression. It also provides a quick check because every row is symmetric, and the sum of the entries in row $n$ is $2^n$.

Connection to Binomial Expansion

The binomial theorem states

$$(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r.$$

The entries in row $n$ are exactly $\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}$. Therefore, row gives

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

Notice that the power of $a$ decreases from $4$ to $0$, while the power of $b$ increases from $0$ to $4$. Every term has total degree $4$.

The coefficient $\binom{n}{r}$ counts the number of ways to choose $r$ factors from which the $b$-term is taken. This explains why the coefficient of $a^{n-r}b^r$ is , and why Pascal's triangle is symmetric: .

For example, using row $3$,

$$(2x-1)^3=1(2x)^3+3(2x)^2(-1)+3(2x)(-1)^2+(-1)^3 =8x^3-12x^2+6x-1.$$

The coefficients are still $1,3,3,1$; the alternating signs come from the powers of $-1$, not from Pascal's triangle.

A common misconception is that row $4$ is the fifth row only because of inconsistent counting. In IB notation, $\mathbb{N}$ includes $0$, so the top $1$ is row $0$.

Exam Technique

For a command such as expand, write every term and check the coefficient pattern, powers, and signs. Verify that there are $n+1$ terms, each term's exponents add to $n$, and the first and last terms are $a^n$ and $b^n$.