Intuition for expansion of higher powers

Question

Consider the expansion of $(x+y)^3$. Without using the binomial expansion we can rewrite this as $(x+y)(x+y)(x+y)$. When we expand it as such, why must we multiply every combination out? For example, we must do $xxx+xxy+xyx+xyy...$. It seems counterintuitive to me for why this is the case. For higher powers, we still multiply out all the combinations, is there an intuitive reason or proof for why we must expand brackets as such?

Answer 1

(I am assuming the knowledge of binomial coefficients, given the question has its tag)

Consider $(x + y)^n$. Suppose we want to expand this. Write this as

$$ (x+y)(x+y)^{n-1} = x(x+y)^{n-1} + y(x+y)^{n-1}$$

Now again write this as

$$ x(x+y)^{n-1} + y(x+y)^{n-1} = x(x+y)(x+y)^{n-2} + x(x+y)(x+y)^{n-2} = x^2(x+y)^{n-2} + xy(x+y)^{n-2} + yx(x+y)^{n-2} + y^2(x+y)^{n-2}$$

Perhaps you might have a feel of where this is going to go. When we are expanding this particular expression, each step gives us double the number of terms of the previous. If we keep on doing this we get a total of $2^n$ terms. This particular number of terms requires us to consider all the possible combinations, if you understand from binomial coefficients, then the sum of all possible combinations from 0 to n is given to be $2^n$ (obtained by setting $x = y = 1$ in our expression) which is equivalent to the number of terms obtained in the expansion.