Binomial Expansion and Pascal's Triangle

The Binomial Theorem

$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

The coefficients $\binom{n}{k}$ come from Pascal's triangle.

Step-by-step: Expand $(x + y)^4$

Pascal's row 4: $1, 4, 6, 4, 1$.

$(x+y)^4 = 1 \cdot x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + 1 \cdot y^4$

Check: Set $x = y = 1$: $(1+1)^4 = 16$. Sum of coefficients: $1+4+6+4+1 = 16$ ✓.

Pascal's triangle pattern

Each number is the sum of the two numbers above it. Row $n$ gives the coefficients for $(x+y)^n$.

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.

Common mistake

When expanding $(2x + 3)^n$ instead of $(x + y)^n$: the $2$ and $3$ get raised to powers too! $(2x)^3 = 8x^3$, not $2x^3$.

Try it in the visualization

Adjust $n$. Pascal's triangle highlights the relevant row. Each term is computed with the coefficient bar chart showing relative sizes. The coefficients $\binom{n}{k}$ form Pascal's triangle. Row $n=4$: $1, 4, 6, 4, 1$.