Let (R) denote the set of all tournaments with score vector R = (r1, r2,…, rn). R. A. Brualdi and Li Qiao (“Proceedings of the Silver Jubilee Conference in Combinatorics at Waterloo,” in press) conjectured that if R is strong with r1 ≤ r2 ≤ … ≤ rn, then |(R)| ≥ 2n−2 with equality if and only if R = (1, 1, 2,…, n − 3, n − 2, n − 2). In this paper their conjecture is proved, and this result is used to establish a lower bound on the cardinality of (R) for every R.