SAMPLE Exam 3 - Fall 2006 - Discrete Structures
1. How many 8-bit ASCII words have at least six 1’s?
2. How many ways can a teacher choose 10 students from a class of 18 Boys and 21 Girls, if she must
choose no more than 2 boys?
3. How many orderings are there of the letters of the word COMPUTERSCIENTIST ?
4. How many ways can I seat 12 people around a circular table?
5. How many ways can I fill a box of 80 chocolates from 20 types if I must have at least 2 of each type in
the box?
6. Let R be the relation on Σ = {0,1} given by R = {(a,b) | a,b ∈ Γ3 and length(a) = length(b)}. Show that
R is an Equivalence Relation.
7. Describe the partition of Γ3 induced by R.
8. Graph S, the relation on {0,1,2,3,4,5,6} given as S = {(a,b) | a + b > 7}.
9. How many ordered pairs are there in the Equivalence Relation, R, that induces the partition:
{ {1,2,3,4,5}, {6}, {7,8,9,0} } of {0,1,2,3,4,5,6,7,8,9}?
10. Find the Primary Keys and P1,2,5 for the trumpet database:
Make Model Year Serial No. Key Finish
Conn 38A 1955 105523 Bb Silver
Conn 40B 1938 52234 Bb Brass
Conn 22A 1966 212203 C Silver
Olds Super 1968 524366 Bb Silver
Olds Super 1952 161230 C Gold
Olds Mendez 1969 210658 Bb Silver
Selmer Radial 1953 14522 Bb Gold
Selmer Paris 1959 18502 Bb Silver
Selmer Radial 1974 64299 Bb Brass
11. What is the probability that a 5-bit integer is prime?
12. What is the probability that a 5-bit integer is prime, given the first (most-significant) bit is 0?
13. Determine whether or not a 5-bit integer is prime is independent of its first bit being 0?
14. Find the Disjunctive Normal Form for a 3-way switch that lights a bulb and is configured so that
when all 3 switched are ON, the bulb is OFF.
15. Find the Disjunctive Normal Form of the polynomial with Truth Table:
x y z F(x,y,z)
1 1 1 0
1 1 0 1
1 0 1 1
1 0 0 0
0 1 1 1
0 1 0 1
0 0 1 0
0 0 0 1
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