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2 problems match “set partitions”. Clear

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1. Partition of a set with size conditions ●●●○○ 50 ptsLet $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$, $\mathcal{A} \cap \mathcal{B} = \emptyset$, the number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$, and the number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$. Find $N$.
2. Placing plants under colored lamps ●●●○○ 50 ptsRachel has two identical basil plants and one aloe plant. She has two identical white lamps and two identical red lamps. Each plant is placed under exactly one lamp, and more than one plant may share a lamp. How many ways are there for Rachel to place her plants under her lamps?