- Apply the Binomial Theorem
A polynomial with two terms is called a binomial. We already know how to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming.
The Binomial Theorem is a powerful tool in counting because it allows us to expand expressions raised to a power, which can then be used to solve problems involving combinations. Specifically, it helps in counting the number of ways to choose a subset of items from a larger set when the order doesn’t matter.
Let’s discuss a shortcut way that will allow us to find [latex](x+y)^n[/latex] without multiplying the binomial by itself [latex]n[/latex] times!
Binomial Coefficients
Have you ever noticed that there’s a pattern to the coefficients when you expand [latex](x+y)^n[/latex]?
These coefficients, known as binomial coefficients, follow a specific pattern that appears in Pascal’s Triangle. Each coefficient represents the number of ways to choose a certain number of terms from the binomial expansion.
binomial coefficients
If [latex]n[/latex] and [latex]r[/latex] are integers greater than or equal to [latex]0[/latex] with [latex]n\ge r[/latex], then the binomial coefficient is
[latex]\left(\begin{gathered}n\\ r\end{gathered}\right)=C\left(n,r\right)=\dfrac{n!}{r!\left(n-r\right)!}[/latex]
- [latex]\left(\begin{gathered}5\\ 3\end{gathered}\right)[/latex]
- [latex]\left(\begin{gathered}9\\ 2\end{gathered}\right)[/latex]
- [latex]\left(\begin{gathered}9\\ 7\end{gathered}\right)[/latex]