In the collector's problem with group drawings, one run draws a uniformly random q-subset of an n-element set, all q coupons being kept, and T denotes the number of runs needed to complete the set. The distribution of T and all of its factorial moments are due to Stadje (1990); we do not claim them, and we say explicitly where our formulas reproduce his -- in particular his relation (2.15) at p = 2 already yields the second moment, so item (i) below is a consolidation rather than a new formula. The asymptotics of E[T] as n -> infinity with q fixed are due to Berend and Sher (2026) and, in a sharper form, to Doumas and Spektor (2026); we do not claim those either, and Section 8 downgrades our own expectation expansion to an independent rediscovery accordingly. What we add is fourfold, of which the first is a consolidation rather than a new formula.(i) An exact closed form for the second moment, hence for the variance, obtained by summing the inclusion-exclusion tail in closed form, together with a uniform one-line derivation of every higher moment. Having now read Stadje (1990) in full we record that his general factorial-moment relation (his eq. (2.15)), specialised to A = S and p = 2, is equivalent to our second-moment formula; item (i) is therefore an explicit specialisation of Stadje's result and is not claimed as new.(ii) Two subtraction-free recursions -- a backward recursion for E[T] in O(nq) time and O(q) memory, and a forward recursion for the whole distribution -- which keep full double-precision accuracy up to n = 1000, whereas the inclusion-exclusion closed form loses all significant digits from n about 40 onwards: at (n,q) = (60,5) its relative error is 4.9 x 10^{-4} and at (80,5) it returns 1.99 x 10^{6} for the true value 77.836.(iii) A quantitative explanation: the relative error of the closed form is predicted by eps*kappa, where eps is the unit roundoff and kappa = sum_i |t_i| / |sum_i t_i| is the condition number of the alternating sum; measured and predicted errors agree to within two orders of magnitude over twenty-six decades.(iv) Asymptotics of the variance, with a complete proof. Writing A = H_n - H_{n-q}, Sigma_2 = sum_{j 0,hence the n log n formVar[T] = (pi^2/6)(n/q)^2 - (n/q^2)(log n + gamma) - ( ((pi^2/6)(q-1) + 1)/q^2 ) n + ((q-1)/(2q^2)) log n + c_0(q) + O(log n / n).For q = 1 everything degenerates exactly to the classical Var[T] = n^2 H_n^{(2)} - n H_n. Doumas and Spektor obtain for the variance only the leading term (pi^2/6)(n/q)^2 (1 + o(1)), so everything beyond it — the n log n term, the n-coefficient, the constant, the closed form and the exponential rate -- is new for q >= 2 as far as we have been able to determine; at q = 1 the expansion collapses to the known result of Doumas and Papanicolaou (2012). We also record the exact support [ceil(n/q), infinity) and point out that the widely quoted formula of Sharif-Hassibi and Xu-Tang belongs to a different model (draw d, keep one), which is why it gives different numbers.