APMO

Let $M$ be a subset of the set $\{1,2,\dots ,1998\}$ and let $|M|=k$. Then the set $M$ can be obtained as the union of $t$ sets $A_1,A_2,\dots ,A_t$ in $(2^t-1{)}^k$ different ways since each element $x\in M$ can belong to $2^t-1$ nonempty families of subsets $A_1,A_2,\dots ,A_t$.

Thus we have $$\sum _{(A_1,A_2,\dots ,A_t)\in F}|A_1\cup A_2\cup \dots \cup A_t|=\sum _{k=1}^{1998}k\left(\genfrac{}{}{0ex}{}{1998}{k}\right)(2^t-1{)}^k$$

$$=1998(2^t-1)\sum _{k=0}^{1997}\left(\genfrac{}{}{0ex}{}{1997}{k}\right)(2^t-1{)}^k=1998(2^t-1)2^{1997t}.$$