How can Pascal’s triangle help expand (a + b) raised to a power?

Pascal’s triangle provides a visual method to determine the numerical coefficients used in the expansion of a binomial expression, such as $(a+b{)}^n$.

To use the triangle for an expansion:
* Identify the row: Locate the row in the triangle that corresponds to the exponent $n$. Counting begins with the top row as $n=0$.
* Extract the coefficients: The numbers in that specific row are the coefficients for the terms in the expansion.
* Apply the variables: As you move across the terms of the expansion, the power of $a$ decreases from $n$ to 0, while the power of $b$ increases from 0 to $n$.

For example, to expand $(a+b{)}^3$, you look at the fourth row (where $n=3$), which contains the numbers 1, 3, 3, 1. Applying these to the variables results in:
$$1a^3{b}^0+3a^2{b}^1+3a^1{b}^2+1a^0{b}^3=a^3+3a^{2}b+3a{b}^2+b^3$$

Each interior entry in the triangle is calculated by adding the two numbers directly above it, reflecting the underlying binomial coefficients defined by the binomial theorem.123