A conjecture on the number of SDRs of a (t,n)-family

A system of distinct representatives (SDR) of a family F=(A₁,...,An) is a sequence (x₁,...,x_n) of n distinct elements with x_i ∈ A_i for 1 ≤ i ≤ n. Let N(F) denote the number of SDRs of a family F; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer t, a family F=(A₁,...,A_n) is called a (t,n)-family if the union of any k1 sets in the family contains at least k+t elements. The famous Hall's theorem says that N(F)1 if and only if F is a (0,n)-family. Denote by M(t,n) the minimum number of SDRs in a (t,n)-family. The problem of determining M(t,n) and those families containing exactly M(t,n) SDRs was first raised by Chang [G.J. Chang, On the number of SDR of a (t,n)-family, European J. Combin. 10 (1989) 231-234]. He solved the cases when 0t2 and gave a conjecture for t3. In this paper, we solve the conjecture.