离散数学Midterm代考 MATH2302/7308代写 离散数学代考

MATH2302/7308 Practice Midterm

离散数学Midterm代考 1. (10 marks total) Note that for this question your answers must be given as a simple expression or a number, not in terms of a

1. (10 marks total)

Note that for this question your answers must be given as a simple expression or a number, not in terms of a generating function.

Alvin, Berry, Cecile and Diana all go apple picking together one weekend.

(a) (5 marks) On Saturday, they pick all 30 apples off an apple tree. How many different ways can the 30 apples be distributed among the four friends?

(b) (5 marks) On Sunday, they pick apples again. At the end, Cecile has at least two more apples than Diana, Berry has at least 3 more apples than Cecile, and Alvin has at least 7 more apples than anybody else. What is the probability of us correctly guessing how many apples they each have, if we also know that Alvin has less than 40 apples?

2. (10 marks total) 离散数学Midterm代考

Note that for this question your answers must be given as a simple expression or a number, not in terms of a generating function.

We wish to rearrange the letters in the word COMBINATORICS.

(a) (5 marks) Determine the number of arrangements that start with a vowel. (Start with A, I or O.)

(b) (5 marks) Determine the number of arrangements that start with a vowel or contain a double C.

3. (10 marks)

Consider the poset based on the set

S = {(3, 1),(4, 2),(7, 3),(6, 4),(8, 5),(9, 6),(10, 7),(5, 8),(27, 9),(4, 10),(1, 11),(13, 12)}

and the binary relation ρ where (a, b)ρ(c, d) if and only if a ≤ c and b ≤ d both hold. (You do not have to check that this is a poset.)

Prove that the size of the maximum antichain in this poset is 5.

4. (10 marks total) 离散数学Midterm代考

Recall that the group of symmetries of a regular triangle id the dihedral group D₃ which has six elements: {I, R₁₂₀, R₂₄₀, F₁, F₂, F₃}. (The identity, rotation around the centre by 120 or 240 degrees, and reflection through one of the three lines connecting a vertex with the midpoint of the opposite side.)

We wish to make triangular panes using coloured glass. These will have the shape of a regular triangle, divided into four equal, smaller, regular triangles. For each smaller triangle we may use

blue or green glass. (The glass is the same colour on both sides.) The picture below shows the outline of such a pane, and one example, where the triangle in the centre and the one in the top corner are blue, and the other two are green. We refer to this picture as A.A

The group D₃ acts on the set of all such triangular pictures. The following questions concern this group action.

(a) (2 marks) Determine |orb(A)|.

(b) (2 marks) Determine |Stab(A)|.

(c) (6 marks) Two triangular panes are considered the same if there is a symmetry in the group D3 that maps the picture of one to the other. Use the counting theorem to determine the number of different triangular panes that can be made.

5. (10 marks)

Find a closed formula for a_n if a₀ = 3, a₁ = 2 and a_i+2 = a_i+1 + 3a_i .

6. (10 marks total) 离散数学Midterm代考

Let s_n be the number of ordered selections of n letters from the set {X, Y, Z} where X is chosen at least once, Y is chosen at least twice, and Z is chosen an even number of times.

(a) (5 marks) Determine the exponential generating function of the sequence (s_n)_n≥0.

(b) (5 marks) Determine a closed formula for the term sn of the sequence (s_n)_n≥0.

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