How does the binomial theorem expand (a + b) to the fourth power?

The binomial theorem provides a systematic way to expand expressions of the form $(a+b{)}^n$. For the fourth power, where $n=4$, the expansion is expressed as the sum of $n+1$ terms, resulting in the following polynomial:

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

The numerical coefficients—1, 4, 6, 4, 1—are known as binomial coefficients. These values can be determined using the combination formula $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{(n-k)!k!}$ or by referencing the fifth row of Pascal’s triangle (where the top row is $n=0$). In combinatorics, these coefficients represent the number of ways to choose $k$ objects from a set of $n$ objects without regard to order.12