Discrete Mathematics Answers

A group of 21 students participates in a discrete mathematics competition. There are Q questions that have to be answered. For each question exactly 4 students are assigned to work together, while the others take a short break. The questions are distributed to ensure that every pair of students only works together on exactly one exercise. This works out exactly!

How many exercises does the competition have?

(Hint: Consider how many pairs occur in a group of 4 students.)

Draw the De Bruijn graph for 2-tuples with digits 0,1,2, and 3.
Make sure to add matching labels to all vertices and edges.

An orientation of a graph G = (V,E) is any directed graph G'= (V,E) arising by replacing each edge {u, v} belonging to E, by the directed edge (u, v) or by the directed edge (v, u).
Show that for every planar graph there is an orientation such that each vertex has at most five outgoing edges. (proof by induction)

For discrete structures there are n exams to check and there are k graders. To guarantee a high quality of grading, every exam may be checked by any number of graders (but always at least by one grader). This means that summed all together, the graders may make up to k*n exam checks. To avoid this, it is required that for each pair of graders there is at most 1 exam that they have both checked. Prove that this rule creates a much better bound of at most ((k+n)^(3/2)+(k+n))/2 exam checks.

Let P = {1,2,3,4} and R = {(1,1), (2,1), (1,2), (2,2), (3,2), (3,4), (4,3), (4,4)} which is a relation on P. Represent this relation as a directed graph. Check whether this relation is an equivalence relation or not.

Draw the Hasse diagram for (i) Pi = {1, 2, 3, 4, 12} and < is a relation such that x < y if x divides y.

How to prove that for all real numbers a,b and c that a² + 5b² + c² ≥ 4ab + 2bc?

How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of a particular suit are chosen

Omar plays a game with his brother Ali, Omar read 4 positive numbers. Ali must find the largest difference between numbers.

Can you help Ali to find the result?
You should use max() predefined function.