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A group of students is selected from a class. Every student in the class is either in the 3rd grade or the 4th grade, and no student is in both the 3rd and the 4th grades. How many students must be selected in order to guarantee that at least five 3rd graders or at least five 4th graders are selected?  

How many solutions are there to the equation  x1 + x2 + x3 + x4 + x5 + x6 = 37, where each xi is an integer that satisfies xi > 1 ?  

License plate numbers in a certain state consists of seven characters. The first character is a digit (0 through 9). The next four characters are capital letters (A through Z) and the last two characters are digits. Therefore, a license plate number in this state can be any string of the form:  Digit-Letter-Letter-Letter-Letter-Digit-Digit 1. How many different license plate numbers are possible?(a) 103  . 264(b) 10 . 9 . 8 . 264(c) 10 . 9 . 8 . 26 . 25 . 24 . 23(d) none of the above [Q1] 2. How many license plate numbers are possible if no digit appears more than once? (a) 103  . 264(b) 10 . 9 . 8 . 264(c) 10 . 9 . 8 . 26 . 25 . 24 . 23(d) none of the above [Q2] 3. How many license plate numbers are possible if no digit or letter appears more than once?(a) 103  . 264(b) 10 . 9 . 8 . 264(c) 10 . 9 . 8 . 26 . 25 . 24 . 23(d) none of the above [Q3]    

1. Which of the following graphs have a Hamiltonian path but not a Hamiltonian circuit? (a) K34(b) Q35(c) K5,6(d) K6,6(e) W35(f) none of them [Q1] 2. Which of the following graphs have both an Eulerian circuit and a Hamiltonian circuit? (a) K34(b) Q35(c) K2,3(d) K6,8(e) Q24(f) none of them [Q2]  

Consider the following claim and the statements Claim: For every real number x, if 0 ≤ x ≤ 3, then 15 – 8x + x2 > 0 (a) 0 ≤ x ≤ 3  (b) x < 0 or x > 3 (c) 15 – 8x + x2 > 0 (d) 15 – 8x + x2 ≤ 0 i. What would be the starting assumption in a proof by contrapositive of the statement above? [ContrapositiveAssumption] ii. What would be proven in a proof by contrapositive of the statement, given the assumption? [ContrapositiveConclusion]

Set Theory Questions.   Indicate which of the following statements are true. For any three sets, A, B, and C, if A ⊆ B, then A × C ⊆ B × C.    [Q1] R2 ⊆ R3.   [Q2] A × (B ∪ C) = (A × B) ∩ (A × C) [Q3] (A ∪ B) – (A ∩ B) = A – B. [Q4] A ∩ (B – A) = ∅. [Q5] Cardinality of the power set of the set {∅} is 2. [Q6]

Students are allowed to use 3, 8 1/2 x11  pages, handwritten, typed, or combination of both, front and back for the final exam. 

using full German sentences, write a short essay, between 200 and 300 words, in which you answer the following: what was your opinion of the games? Do you think they are a good way to teach kids to recycle and separate trash? Why or why not? What would you do differently? What worked or did not work?

What is the prefix length notation for the subnet mask 255.255.255.224?  

An employee at a branch office is creating a quote for a customer. In order to do this, the employee needs to access confidential pricing information from internal servers at the Head Office. What type of network would the employee access?