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109年 - 109 教育部公費留學考試：離散數學#125735
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題組內容
1. Prove the following two statements.
(a)
相關申論題
(b) Use the formula from (a) to prove
#534242
(a) What is the coefficient of in the expansion of ?
#534243
(b) Use the result of (a) to compute the coefficient of in the expansion of .
#534244
3. Prove or disprove the following statement. Given a bipartite graph, if the number of vertices of odd degree on the left subgraph (subgraph A below) is odd, then the number of vertices of odd degree on the right subgraph (subgraph B below) is also odd.
#534245
4. Simplify the Boolean formula tr(M²) given the matrix M below with as few operations (AND, OR) as possible. The trace of a square matrix, denoted as tr(*) is defined by the summation of all diagonal elements in the matrix.
#534246
5. Convert the following finite state machine to another machine where every two transitions becomes one. That is, we have two alphabets, one after another in each transition in the new machine while all the states remain to be the same to the original machine. We assume the starting state is A.
#534247
1. Prove that the product of two consecutive positive integers is always even.
#534248
2. Compute where n can be even or odd.
#534249
3. How many different possible shortest lattice paths to go from point (0,0) to point (7,5)? No need to give the final answer but showing how to calculate the answer is good enough.
#534250
4. Assume a person may get Covid-19 infected each time this person goes out by the probability p, compute the probability of getting Covid-19 if this person goes out 10 times. We also assume the probability of getting Covid-19 is independent for each going-out trip.
#534251
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