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

Moving one ball can destroy at most one empty bin and create at most one empty bin; the net number of empty bins therefore changes by at most one. Thus $f$ has the bounded differences property with $c_j=1$ for all $m$ ball coordinates. The upper- and lower-tail forms of the McDiarmid inequality give, for $t>0$,