How can we deduce the Binomial Theorem by interpreting the binomial coefficient as the number of subsets with n elements?

Question

I wonder how can I interpret the binomial coefficient by using the interpretation that the binomial coefficient give us the number of subsets with k elements

Answer 1

The Binomial Theorem states that if $x, y \in \mathbb{R}$ and $n$ is a non-negative integer, then $$(x + y)^n = \sum_{k = 0}^n \binom{n}{k}x^{n - k}y^k$$ Each term is formed by selecting either an $x$ or a $y$ from each of the $n$ factors. The coefficient of $x^{n - k}y^{k}$ is the number of ways of selecting a $y$ from $k$ of the $n$ factors.

Answer 2

We have $$ (1+x)^n=\overbrace{(1+x)(1+x)\cdots(1+x)}^{n\text{ factors}} $$ To compute how many terms of $x^k$ are contributed by applying the distributive property, count the number of ways to choose the $k$ factors that contribute an $x$ and the $n-k$ that contribute a $1$ out of the $n$ factors. This gives $$ \binom{n}{k}x^k $$ Summing over the possible values of $k$ gives $$ (1+x)^n=\sum_{k=0}^n\binom{n}{k}x^k $$