三、Please calculate the number of positive integers greater than 1 (>
1) and not exceeding 150 (≤ 150) that are divisible by none of 3,
5, or 7 by using The Principle of Inclusion–Exclusion. You should
explain your answer. (10%)

四、Paul has two coolers. Cooler 1 contains 8 cans of cola and 3 cans of lemonade. Cooler 2 contains 5 cans of cola and 7 cans of lemonade. Paul randomly selects one can from Cooler 1 and transfers it into Cooler 2. Subsequently, Betty randomly selects two cans from Cooler 2. Given that both of Betty's selections are cans of cola, what is the probability that the can Paul initially transferred was lemonade? (10%)

五、Solve the following recurrence relation: ?! + 6?!%# − 16?!%$ = 6, given that ?" = 0 and ?# = 2.

(10%)

六、Determine whether the poset (Partially Ordered Set) (?, ⊆) is a lattice where ? ={∅, {1}, {2}, {4}, {1,2}, {1,4}, {3,4}, {1,2,4}, {2,3,4}, {1,2,3,4}} Please draw its Hasse diagram and explain your answer. (10%)

七、Graph (1) Let ? = (?, ?) be an undirected graph with ? components, where |?| = ? and |?| = ? . Please prove that c ≥ ? − ? . (10%)

(2) Consider the complete bipartite graph ?&,! , where the vertex set is partitioned into two disjoint sets ?# and ?$ with |?# | = ? and |?$ | = ?. Derive the necessary and sufficient conditions for ? and ? such that ?&,! contains an Eulerian path but NO Eulerian circuit. (Consider all possible cases for ? and ? ). (10%)

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