107年 - 107 國立中山大學_碩士班招生考試_資工系(甲組)：離散數學#105787

(a) ((x³-3x)(1-x)⁵)+3x⁷+5.

(b) -2/((x-1)(x-2).

(a) Find the exponential generating function for the above counting such that the coefficient of is the answer of the above counting.

(b) Apply the above exponential generating function to find the answer of the above counting where |A| = 10 and |B| = 3.

(a) For each integer n ≥ 1, let an be the number of strings in A* of length n. Find and solve a recurrence relation for .

(b) For each integer n z 1, let bn be the number of strings in A* which are of length n and exactly divided by 2 when we regard each of the strings as a binary number. Find .

(a) What is the order of 6? Why?

(b) Find all generators of the group.

(a) What is the order of the group? Why?

(b) Find the inverse of 25.

(a)3(4,5)=(4,5)⊕(4,5)⊕(4,5)=?

(b) What is the identity of the group?

(c) What is the inverse of (a, b)?

(d) [8%] Let (G, +) be an additive group and let h: ZxZ → G be a group homomorphism where h(3, 2) = u and h(-2,6) = v. Please express h(18, 22) in terms of u and v.

107年 - 107 國立中山大學_碩士班招生考試_資工系(甲組)：離散數學#105787

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