The Binomial Theorem#
The Binomial Theorem
The expansion of the complex polynomial \((x+y)^n\) is the following:
\[ (x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k = \sum_{k=0}^n \binom{n}{k} x^{k} y^{n-k}, \]
where
\[ \binom{n}{k} \overset{\text{def}}{=} \frac{n!}{k! (n-k)!}. \]
Definition: Binomial Coefficient
The numbers \(\binom{n}{k}\) are known as binomial coefficients.
Proof
TODO
Theorem: Symmetry of Binomial Coefficients
The binomial coefficients have the following property:
\[ \binom{n}{k} = \binom{n}{n-k} \]
Proof
By definition:
\[ \begin{aligned} \binom{n}{k} &= \frac{n!}{k! (n-k)!} \\ \binom{n}{n-k} &= \frac{n!}{(n-k)!(n-(n-k))!} = \frac{n!}{(n-k)!k!} \end{aligned} \]
Theorem: Recursion of Binomial Coefficients
The binomial coefficients have the following recursive property:
\[ \binom{n + 1}{k} = \binom{n}{k} + \binom{n}{k-1} \]
Proof
We write out the right-hand side:
\[ \binom{n}{k} + \binom{n}{k-1} = \frac{n!}{k! (n-k)!} + \frac{n!}{(k-1)!(n-k+1)!} \]
We now use the definition of the factorial:
\[ \begin{aligned} &\frac{n!}{k! (n-k)!} + \frac{n!}{(k-1)!(n-k+1)!} \\ &= \frac{n!}{k(k-1)!(n-k)!} + \frac{n!}{(k-1)!(n-k+1)(n-k)!} \\ &= \frac{n!(n-k+1) + n!k}{k(k-1)!(n-k)(n-k+1)} \\ &= \frac{n!(n+1)}{k!(n+1-k)!} \end{aligned} \]
This is equal to the definition of the left-hand side:
\[ \binom{n+1}{k} = \frac{(n+1)!}{k!(n+1-k)!} \]
Theorem: Sum of Binomial Coefficients
The sum of the binomial coefficients \(\binom{n}{k}\) from \(0\) to \(n\) is \(2^n\):
\[ \sum_{k = 0}^n \binom{n}{k} = 2^n \]
Proof
We use the binomial theorem itself:
\[ (x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k \]
Set \(x = 1\) and \(y = 1\):
\[ \begin{aligned} (1 + 1)^n &= \sum_{k=0}^n \binom{n}{k} \cdot 1^{n-k} \cdot 1^k \\ 2^n &= \sum_{k=0}^n \binom{n}{k} \end{aligned} \]