99年國立高雄第一科技大學電通所離散數學

1.

In how many ways can we distribute n identical objects into k distinct containers so that no container is left empty?

2.

In how many ways can we distribute n identical objects into k distinct containers so that containers may be empty?

3.

In how many ways can we distribute n distinct objects into k identical containers so that no container is left empty?

4.

In how many ways can we distribute n distinct objects into k identical containers so that containers may be empty?

5.

In how many ways can we distribute n distinct objects into k distinct containers so that no container is left empty?

6.

In how many ways can we distribute n distinct objects into k distinct containers so that containers may be empty?

7.

Find all real numbers x such that 

8.

Find all real numbers x such that 

9.

Find all real numbers x such that 

10.

Find all real numbers x such that 

Let n ∈ Z+ where n > 1, and consider the directed graph G = (V, E) with V = {1, 2, 3, …, n} and E = {(i, j) | n ≥ i ≥ j ≥ 1}.

11.

How many edges are there for this graph?

Let n ∈ Z+ where n > 1, and consider the directed graph G = (V, E) with V = {1, 2, 3, …, n} and E = {(i, j) | n ≥ i ≥ j ≥ 1}.

12.

How many directed paths, in which no vertex may be repeated, exist in G from n to 1?

13.

For n ∈ Z⁺, solve a recurrence relation for f(n) of the form f(1) = 1, f(2) = 2, f(n) = 2f(n − 1) + 3f(n − 2), n = 3, 4, 5, 6, ….

14.

For n = 2^k, k ∈ N, solve a recurrence relation for f(n) of the form f(1) = 1, f(2) = 2, f(n) = 2f(n/2) + 3f(n/4), n = 4, 8, 16, 32, ….

15.

In how many ways to arrange the letters in HORSES with no consecutive S's?

16.

In how many ways to arrange the letters in HORSES so that none of the letters H, O, R, E is in its original position and the letter S is not in the fourth or sixth position?

Let the set A = {1, 2, 3, 5, 6}. Define the relation R on A by aRb if |a − b| ≤ 1.

17.

Find R and R², and their relation matrices M(R) and M(R²).

Let the set A = {1, 2, 3, 5, 6}. Define the relation R on A by aRb if |a − b| ≤ 1.

18.

Determine whether the relation R is reflexive, symmetric, anti-symmetric, or transitive.

Let the set A = {1, 2, 3, 5, 6}. Define the relation R on A by aRb if |a − b| ≤ 1.

19.

Determine whether the relation R² is reflexive, symmetric, anti-symmetric, or transitive.

Let  , the generating function for the sequence a₀, a₁, a₂, a₃, …. Now let n ∈ Z⁺, n fixed.

20.

Find the generating function for the sequence 0, 0, …, 0, a₀, a₁, a₂, a₃, …, where there are n leading zeros.

Let  , the generating function for the sequence a₀, a₁, a₂, a₃, …. Now let n ∈ Z⁺, n fixed.

21.

Find the generating function for the sequence 0, a₀, 0, a₁, 0, a₂, 0, a₃, 0, a₄, ….

Let  , the generating function for the sequence a₀, a₁, a₂, a₃, …. Now let n ∈ Z⁺, n fixed.

22.

Find the generating function for the sequence a_n, 0, a_n+1, 0, a_n+2, 0, a_n+3, 0, a_n+4, ….