PART A
1. m mod n will have values ranging from 0 to n.
T or F
2. 42 MOD 8 and 50 MOD 7 are congruent.
T or F
3. The base system of the value 635 must be either decimal or octal.
T or F
4. The value 457 has a maximum of 21 possible prime factors because the square root of 457 is 21+.
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(8,5) = 56; C(9,2) = 72
T or F
7. Consider the following directed relations on {1, 2, 3 } :
R1 = { (1,1), (2,2), (3,3) }, and
R2 = { (1,2), (1,3), (3,2) }.
R1 is reflexive 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 543 is 544.
T or F
9. According to the Pigeonhole principle, when (m+6) items are to be placed in (m+4) 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 88 positive integers not exceeding 423 that are divisible by either 7 or 13.
T or F
14. A brand of shirt comes in four basic colors, has male, female and unisex versions and has five sizes for each. This brand has a maximum of 12 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). -43 MOD 7
B). -92 MOD 8
2. Determine the Base10 expansion of (3DE) 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 10,647. Show the result in proper exponential form.
5. Given:
A = 980
B = 2079
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 (249, 680).
7. Convert (1101 0101) Base2 to:
A). ( ) Base16
B). ( ) Base10
8. Given 3526BASE10. Determine the equivalent value in BASE5.
Hint: Use the Euclidean Algorithm
9. Define: (show intermediate work)
A. P(8,6) =
B. C(9,3) =
10. What is the coefficient of ( x^4 y^3 ) in the expansion (x - 2y)^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 six fair coins are flipped eight times. What is the probability that each result has three heads in each flip?
13. f(n)= 3*f(n/2) - 4 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 6432 are divisible by neither 15 nor 18?
15. Given |A| = |B| = |C| = 60, |A INT B| = 15, |B INT C| = 30, |A INT B INT C| = 5, and
|A UNION B UNION C| = 120 elements.
|A INT C| = ?
16. List the next SIX terms of the lexicographic ordering of the n-tuple 32654 where each digit is in the set {2,3,4,5,6}.
17. Which lottery presents the player with the best odds for winning, (A or B)? Defend your
answer.
A = C(39,4)
B = C(40,4)
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.
.......................................................
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 Base5 value of 1534Base8. 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. Note that the first two letters cannot repeat.
What is the maximum possible number of station call signs? Defend your answer.