113年 - 113 教育部公費留學考試試題：離散數學#125718

1. What is the probability that none of the n couples pair with their spouse as n goes to infinity? (20%)

2. You have two copies of letter A and an unlimited supply of letters B, C and D. How many words of length 9 can you make from this such that all A's are used, the fifth letter is an A, and there is no B appearing between A's. (20%)

3. Eight problems are distributed to four students, each getting at least one problem. Two problems are easy and therefore cannot be distributed to the same student. How many ways are the distributions of the eight problems to the four students? (15%)

4. Let G=(V,E) and H=(V,F) be simple undirected graphs, where V is the vertex set, and E and F are the edge sets. The union of E and F forms the edge set of the complete graph on V. Prove or disprove the following statement: One of G or H must be connected. (15%)

5. is the sequence generated by List out the first five terms of the sequence generated by . (15%)

6. How many words can be made by rearranging aaabbccdd, such that no 'a' appears somewhere to the right of some 'c'. (15%)