Dress-up time

Let $A_j$ be the set of dressing orders that do not use the $j$ th dress, for $1 \le j \le 3$ . Then $\begin{aligned} |A_1| = |A_2| = |A_3| & = \; 2^7 \\ |A_1 \cap A_2| = |A_1 \cap A_3| = |A_2 \cap A_3| & =\; 1 \\ |A_1 \cap A_2 \cap A_3| & = \; 0 \end{aligned}$ and hence $|A_1 \cup A_2 \cup A_3| \; = \; 3 \times 2^7 - 3 \times 1 + 0$ so that $|(A_1 \cup A_2 \cup A_3)'| \; = \; 3^7 - 3\times 2^7 + 3 \; = \; \boxed{1806}$ using the Inclusion-Exclusion Principle.