MHB How Does the Constant '2' Arise in Binomial Theorem Equalities?

[gladeligen]

View attachment 6378

Hi!

My first post!

I was wondering if anyone could explain this equality in words?
I do not understand how the k became a constant (2) on the right side?Best
Flippa

 

[Evgeny.Makarov]

Hi, and welcome to the forum!

Proving equalities in words is not something mathematicians usually do. They usually prove equalities by a series of formulas. Perhaps the most intuitive type of proofs of similar facts is combinatorial proofs, where you consider a set and count its size in two different ways. You can find one such proof on StackExchange. It relies on the fact that $\sum_{k=0}^n\binom{n}{k}=2^n$. This fact also has a combinatorial proof: $\binom{n}{k}$ is the number of subsets of size $k$ of the set of size $n$, so the left-hand side is the number of all subsets of a set of size $n$. This number is known to be $2^n$ because each of the $n$ elements either occurs in a subset or does not (thus $n$ consecutive independent yes/no choices). There are many other proofs of the same fact on the SE page and pages linked therein: some use induction, others differentiation, etc.