Real Binomial Expansions
Result
Suppose $x$ and $y$ are real numbers.
For any natural number $n$,
\[ (x+y)^n = \sum_{k = 0}^{n} {n \choose k} x^{k}y^{n-k} \]
Roughly speaking, expand $(x+y)^n$ using the
distributive law, and count the number of terms
containing $k$ $x$’s and $n-k$ $y$’s.
The number will be same as the number of
ways of choosing $k$ elements out of $n$,
which is ${n \choose k}$.
Each such term contributes $x^ky^{n-k}$ to the
sum.
Future editions will include a full proof, and
perhaps some visualization.
The expression $x+y$ is called a binomial and this result is often called the binomial formula, binomial theorem, binomial identity. For these reasons, ${n \choose k}$ is often called a binomial coefficient.