HMMT November 2020 T10

Let $x$ and $y$ be non-negative real numbers that sum to 1 . Compute the number of ordered pairs $(a, b)$ with $a, b \in\{0,1,2,3,4\}$ such that the expression $x^{a} y^{b}+y^{a} x^{b}$ has maximum value $2^{1-a-b}$.