On the number of a SDRs of a valued (t,n)-family

A system of distinct representatives (SDR) of a family $F = (A_1, \cdots, A_n)$isasequence$(x_1, \cdots, x_n)$of$n$distinctelementswith$x_i \in A_i$for$1 \le i \le n$\mathrm{.}Let$N(F)$denotethenumberofSDRsofafamily$F$; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer $t$, a family $F=(A_1,\cdots ,A_n)$ is called a $(t,n)$-family if the union of any $k\ge 1$ sets in the family contains at least $k+t$ elements. The famous Hall’s Theorem says that $N(F)\ge 1$ifandonlyif$F$isa$(0,n)$-family\mathrm{.}Denoteby$M(t,n)$ 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 [European J. Combin.{\bf 10}(1989), 231-234]. He solved the cases when $0\le t\le 2$andgaveaconjecturefor$t\ge 3$. In this paper, we solve the conjecture. In fact, we get a more general result for so-called valued $(t,n)$-family.