general formula in a binomial expansion

Question

Is it true that the binomial expansion of ${(x-y)}^n$ is the same as the binomial expansion of ${(y-x)}^n$ if $n$ is even but is not the same if $n$ is odd ?

Answer 1

One easy thing to see is that $$ (x-y)^n = \left[-(y-x)\right]^n = (-1)^n (y-x)^n. $$ Can you finish this from here?

Answer 2

indeed, we have for $n=2$ $$(x-y)^2=x^2-2xy+y^2=(y-x)^2=y^2-2xy+x^2$$ and for $n=3$ $$(x-y)^3=x^3-3x^2y+3xy^2-y^3$$ and $$(y-x)^3=y^3-3y^2x+3yx^2-x^3$$