PART A
1. m mod n will have values ranging from 0 to n+1.
2. 44 MOD 7 and 65 MOD 9 are congruent.
3. A Permutation of the elements of a set is an ordered arrangement of the elements of the set.
4. The value 739 has a maximum of 27 possible prime factors because the square root of 739 is 27+.
5. The base system of the value 627 must be either decimal or octal.
6. P(8,3) = 56; P(9,4) = 3024
7. Consider the following relations on {1, 2, 3 } :
R1 = { (1,1), (2,2), (3,3) }, and
R2 = { (1,2), (1,3), (2,3) }.
R1 is reflexive and R2 is transitive
8. Using members of the set {1, 3, 4, 5, 7, 8, 9}, the next larger P(7,3) permutation after 453 is 454.
9. According to the Pigeonhole principle, when (m+5) items are to be placed in (m+7) boxes, there will be more than one item in at least one box.
10. Pascal's Triangle yields the value of the coefficients of an algebraic expansion.
11. The probability of picking a "face" card (Jack, Queen or King) from a standard deck of playing cards is C(52,12).
12. P(n,r) is equal to or greater than C(n,r) when n => 1.
13. There are 124 positive integers not exceeding 562 that are divisible by either 7 or 11.
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.
PART B
1. Determine:
A). -43 MOD 4
B). -79 MOD 6
2. Determine the Base10 expansion of (B5E) 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 47432. Show the result in proper exponential form.
5. Given:
A = 4410
B = 5500
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 (382347, 83853).
7. Convert (1011 1101) Base2 to:
A). ( ) Base16
B). ( ) Base10
8. Given 4637BASE10. Determine the equivalent value in BASE5.
Hint: Use the Euclidean Algorithm
9. Define: (show intermediate work)
A. P(8,6) =
B. C(7,8) =
10. What is the coefficient of ( x^5 y^2 ) in the expansion (2x - y)^7 ? You may leave the answer in a proper intermediate form.
11. Each locker in a building is labeled with two upper-case alpha characters followed by four 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 five times. What is the probability that each result has four heads in each flip?
13. f(n)= 3*f(n/2) + 6 when n is even and f(1) = -2.
a. What is the value of f(4)?
b. What is the value of f(8)?
14. How many positive integers not exceeding 8154 are divisible by neither 9 nor 12?
15. Given |A| = |B| = |C| = 90, |A INT B| = 25,
|A INT C| = 35, |A INT B INT C| = 15, and
|A UNION B UNION C| = 185 elements.
|B INT C| = ?
16. List the next SIX terms of the lexicographic ordering of the n-tuple 32576 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(42,3)
B = C(44,2)
18. Determine if the following zero-one matrix is:
a. reflexive | 1 0 1 |
b. symmetric | 1 0 0 |
c. transitive | 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. 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.