When are subset sums equidistributed modulo m?

For a triple (n, t,m) of positive integers, we attach to each t-subset S = {a1, . . . , at} ⊆ {1, . . . , n} the sum f(S) = a1 + · · · + at (modulo m). We ask: for which triples (n, t,m) are the ( n t ) values of f(S) uniformly distributed in the residue classes mod m? The obvious necessary condition, that m divides ( n t ) , is not sufficient, but a q-analogue of that condition is both necessary and sufficient, namely: q − 1 q − 1 divides the Gaussian polynomial ( n t )