# Proof of the Binomial Theorem

Let's recall what the binomial theorem states. The theorem gives us a formula for calculating the expansion of a binomial made up of first degree monomials (i.e. expansion of $$(x+y)^n$$). What's a binomial or a monomial for that matter? Let's start with a monomial. A monomial is an expression like $$y^2$$; more complex monomials can be written, but they won't apply to this theorem. A first degree monomial is anything like $$x,y,z$$. Easy, right? Pascal appears rather celebrated in this domain. He has something called Pascal's triangle named after him. To avoid getting sidetracked I will refer the interested reader to Wikipedia for a great discussion on Pascal's triangle. https://en.wikipedia.org/wiki/Pascal's_triangle

Proof of the Binomial Theorem

The proof is based on induction. Just recall when doing proofs, that involve proving something is true up to some variable $$n$$, induction is your friend. Hence, this proof uses induction as well. So let's start. We want to prove that,

$(x+y)^n=\sum_{j=0}^n {n \choose j} x^jy^{n-j}$ Where, ${n \choose j}=\frac{n!}{j!(n-j)!}$ Proof by induction, for $$n=1$$, $(x+y)^1=\sum_{j=0}^1 {1 \choose j} x^1 y^{1-j}=y+x$ Which is true. So, now assume true for $$n$$ and show it's true for $$n+1$$, \begin{align*} (x+y)^n&=\sum_{j=0}^n {n \choose j} x^j y^{n-j}\\ (x+y)(x+y)^n&=(x+y)\sum_{j=0}^n {n \choose j} x^j y^{n-j}\\ (x+y)^{n+1}&=\sum_{j=0}^n {n \choose j}x^{j+1} y^{n-j}+\sum_{j=0}^n {n \choose j} x^j y^{n-j+1} \end{align*} Let $$k=j+1$$ which $$\Rightarrow j=k-1$$ so therefore $$n-j=n-(k-1)$$ or $$n-j=n-k+1$$ \begin{align*} &=\sum_{k=1}^{n+1} {n \choose {k-1}}x^k y^{n-k+1}+\sum_{k=0}^n x^k y^{n-k+1}\\ &=\sum_{k=1}^n\left[{n \choose {k-1}}+{n\choose k}\right]x^k y^{n-k+1}+y^{n+1}+x^{n+1} \end{align*} by Pascal's rule (see proof at bottom of post) we have, $=\sum_{k=0}^{n+1}{{n+1}\choose k} x^k y^{n+1-k}$ So this proves it is true for $$n+1$$, hence we have, $(x+y)^n=\sum_{j=0}^n {n\choose j}x^jy^{n-j}$

Whew, that was non-trivial as they say.

If you want to know more about the Binomial Theorem, consider checking out the Wikipedia article,

http://en.wikipedia.org/wiki/Binomial_theorem

P.S. Note: the binomial theorem is often our friend when we are doing more advanced math proofs that require expanding on the form $$(x+\Delta x)^n$$, as seen in some calculus proofs.