# Why does binomial expansion specify raising a base to zero or one?

While reviewing the Wikipedia article surrounding Pascal's triangle, I came across the following:

Pascal's triangle determines the coefficients which arise in binomial expansions. For example, consider the expansion:

$$(x + y)^2 =$$

$$x^2 + 2xy + y^2 =$$

$$1x^2y^0 + 2x^1y^1 + 1x^0y^2$$

However, since $$x^0 = 1$$, and, $$x^1 = x$$, why not simply write it as:

$$(x + y)^2 =$$

$$x^2 + 2xy + y^2 =$$

$$x^2 + 2xy + y^2$$

Is it because of the keyword "expansion", or am I missing something critical here?

Edit: Ohhhhhhhh I get it now... Thanks to @JamesA for the answer and pointing out in the comments that I had ported it incorrectly. Had I ported it over correctly, it would've made sense and we wouldn't have this question.

• @JamesA with that edit, it became immediately obvious why it was written that way lol 😂 Sep 15, 2021 at 21:22

The coefficients of $$1$$ are written to make the coefficients match the numbers in Pascal's triangle.
The powers of zero and one make it clearer that the sum of the powers of $$x$$ and $$y$$ in each term are the same. They also make clear a quick way of writing them out, namely, increase the power of $$y$$ by $$1$$ each term and decrease the power of $$x$$ by $$1$$.
They're there more for teaching and understanding. Out in the real world you'll probably never see coefficients or powers of $$1$$ or $$0$$.