Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 3 / c / Solution

Let $B$ be the set of bins that are occupied in configuration $x$ but empty in configuration $y$. For each $b\in B$, choose the lowest-numbered ball $j_b$ lying in $b$ under $x$. Then ${\alpha }_{j_b}(x)=1$ and $x_{j_b}\ne y_{j_b}$. Distinct bins choose distinct balls, soEvery increase in the number of empty bins is accounted for by a newly empty bin, while newly occupied bins only decrease that number. Hence $f(y)-f(x)\le |B|$, proving the stated inequality.