Sample Exam Questions
1. A coin is flipped 10 times; each flip comes up either heads or tails. How many
possible outcomes:
a) are there in total?
b) Contain exactly 2 heads?
c) Contain at least 3 heads?
d) Contain the same number of heads and tails?
2. What is the x5y3 coefficient of the expansion of (10x-2y)8?
3. What is the probability that a five-card poker hand contains cards of five different
kinds (in other words, no pairs, triples, or four-of-a-kinds)?
4. R is a relation on the set of all positive integers, and R={(x,y) | x and y have the
same prime divisors}. Show that R is an equivalence relation.
5. Suppose set A = {1,5,9,13,17}. Suppose R is a relation on A such that
R={(1,1),(1,5),(5,1),(5,5),(9,17)(17,13),(13,9)}
a) Draw the binary matrix that represents R
b) Draw a directed graph that represents R
c) State whether or not the relation is:
 reflexive
 symmetric
 antisymmetric
 transitive
6. Use the graph below to answer questions a-b:
a) Find an Euler circuit in the graph
b) Is the graph bipartite? Why or why not?
c) Find the cut edges and cut vertices in the graph:
Discrete Computational Structures
Sheller
s06samp3 - work
Spring 2006
Page 1
7. Construct a binary expression tree from the following infix expression, then write
the expression in prefix and postfix form:
(p  (p  q))  q
Tree:
Prefix expression:
Postfix expression:
8. Suppose the address of vertex v in an ordered rooted tree is 3.2.4.5.3 Answer
questions a through e concerning the tree to which v belongs
a) At least how many siblings does v have? Why?
b) What is the address of v's parent node?
c) If the tree is a full m-ary tree, what is the value of m? Why?
d) Given your answer in part c, how many internal vertices would the tree have?
e) Given the address of v, and not knowing whether or not the tree is full, what is
the minimum number of vertices in the tree? Explain your answer.
Discrete Computational Structures
Sheller
s06samp3 - work
Spring 2006
Page 2