PART A
1. m mod n will have values ranging from 0 to n-1.
T or F
2. 42 MOD 6 and 49 MOD 7 are congruent.
T or F
3. The base system of the value 375 must be either decimal or octal.
T or F
4. The value 638 has a maximum of 25 possible prime factors because the square root of 638 is 25+.
T or F
5. A Permutation of the elements of a set is an ordered arrangement of the elements of the set.
T or F
6. P(9,2) = 72; C(7,4) = 35
T or F
7. Consider the following relations on {1, 2, 3 } :
R1 = { (1,1), (2,2), (3,3) }, and
R2 = { (1,2), (1,3), (3,2) }
R1 is symmetrical and R2 is transitive
T or F
8. Using members of the set {1, 3, 4, 5, 7, 8}, the next larger P(6,3) permutation after 342 is 343.
T or F
9. According to the Pigeonhole principle, when (m+7) items are to be placed in (m+5) boxes, there will be more than one item in at least one box.
T or F
10. Pascal's Triangle yields the value of the coefficients of an algebraic expansion.
T or F
11. The probability of picking a "face" card (Jack, Queen or King) from a standard deck of playing cards is C(52,12).
T or F
12. P(n,r) is equal to or greater than C(n,r) when n => 1.
T or F
13. There are 101 positive integers not exceeding 457 that are divisible by either 7 or 11.
T or F
14. A brand of shirt comes in three basic colors, has male, female and unisex versions and has five
sizes for each. This brand has a maximum of 11 different varieties.
T or F
PART B
Divided questions are worth 3 points for each section - or as indicated.
SHOW ALL WORK (within reason) in intermediate stages. Clearly identify the final answer.
1. Determine:
A). -51 MOD 6
B). -83 MOD 7
2. Determine the Base10 expansion of (EC4) Base16
3. Define if the each set of integers are mutually relatively prime. Defend your conclusion.
A). {8, 44, 55}
B). {7, 15, 26, 29, 37, 42}
4. Find the prime factors of the value 92,565. Show the result in proper exponential form.
5. Given:
A = 2750
B = 2205
Define by factoring:
A). gcd (A, B) show in exponential form
B). lcm (A, B) show in exponential form
6. Using the Euclidean Algorithm, determine:
GCD (23400, 770).
7. Convert (1010 1011) Base2 to:
A). ( ) Base16
B). ( ) Base10
8. Given 3419BASE10. Determine the equivalent value in BASE5. Hint: Use the Euclidean Algorithm
9. Define: (show intermediate work)
A. P(9,7) =
B. C(11,8) =
10. What is the coefficient of ( x^4 y^3 ) in the expansion (x - 3y)^7 ? You may leave the answer in a proper intermediate form.
11. Each locker in a building is labeled with three upper-case alpha characters followed by two Base 16 characters. What is the maximum number of different locker numbers that can be generated?
12. A group of seven fair coins are flipped five times. What is the probability that each result has three heads in each flip?
13. f(n)= 2*f(n/2) - 5 when n is even and f(1) = -3.
a. What is the value of f(4)?
b. What is the value of f(8)?
14. How many positive integers not exceeding 7235 are divisible by neither 15 nor 12?
15. Given |A| = |B| = |C| = 75, |A INT B| = 20,
|A INT C| = 40, |A INT B INT C| = 10, and
|A UNION B UNION C| = 145 elements.
|B INT C| = ?
16. List the next SIX terms of the lexicographic ordering of the n-tuple 36257 where each digit is in the set {2,3,5,6,7}.
17. Which lottery presents the player with the best odds for winning, (A or B)? Defend your answer.
A = C(37,5)
B = C(38,4)
18. Determine if the following zero-one matrix is:
a. reflexive T or F | 1 1 1 |
b. symmetric T or F | 1 0 0 |
c. transitive T or F | 1 0 1 |
Defend your answers.
.......................................................
OPTIONAL QUESTION DO ONE.
A Develop the Basis Step of the algorithm to determine the number of terms (cardinality) of the union of n mutually intersecting sets. Show your work.
For example, the cardinality of the union of three mutually intersecting sets is
C(3,1) + C(3,2) + C(3,3) = 3+3+1 = 7.
B. Determine the Base3 value of 1642Base8. Show your work
C. In the past, US radio stations had call three or four letter call signs beginning with either K or W. For example: KSO, KDKA, WHO and WINZ. What is the maximum possible number of station call signs? Defend your answer.