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A system of distinct representatives (SDR) of a family $F = (A_1, \cdots,A_n)$ is a sequence $(x_1, \cdots, x_n)$ of $n$ distinct elements with $x_i \inA_i$ for $1 \le i \le n$. Let $N(F)$ denote the number of SDRs of a family $F$;two SDRs are considered distinct if they are different in at least onecomponent. For a nonnegative integer $t$, a family $F=(A_1,\cdots,A_n)$ iscalled a $(t,n)$-family if the union of any $k\ge 1$ sets in the familycontains at least $k+t$ elements. The famous Hall's Theorem says that $N(F)\ge1$ if and only if $F$ is a $(0,n)$-family. Denote by $M(t,n)$ the minimumnumber of SDRs in a $(t,n)$-family. The problem of determining $M(t,n)$ andthose 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 $0\let\le 2$ and gave a conjecture for $t\ge 3$. In this paper, we solve theconjecture. In fact, we get a more general result for so-called valued$(t,n)$-family.

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