MAT243 Discrete Mathematical Structures

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MAT243 Discrete Mathematical Structures

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MAT243 Discrete Mathematical Structures

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Course Code: MAT243
University: Arizona State University

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Country: United States

Question:

A large game developer wishes to test a new game on every possible combination of supported CPU and GPU types, within reason. The assumption is that everyone’s gaming PC will have one CPU and one external GPU (CPU-integrated graphics processors are not used even when they exist). 37 Nvidia and 16 AMD GPUs are supported.Intel and 1 AMD CPU are seen as sufficient to cover the most popular CPU choices. The game is released for a single operating system and all other hardware variations are ignored. On how many different systems does the game need to be tested? Explain your answer by referring to the relevant rules we covered.
At the Phoenix annual meeting of the pet owner’s club, a show of hands determined that 41 people in attendance owned one or more dogs but no cat; 38 owned one or more cat but no dogs; and 12 owned both types of pet.
A dog/cat owner committee of 6 people is to be formed. It is thought that the diversity of the dog/cat owning community is best represented if 2 people on that committee only own a dog or dogs; 2 people only own a cat or cats; and 2 people own both dog(s) and cat(s). In how many ways can this committee be selected? Explain your reasoning and show work.
How does the solution to the previous problem change if the two people selected out of each of the three distinct groups do not have equal power on the committee, but rather, the first selected is the “speaker” for that group, and the second selected is just an advisor? Explain your answer
(a) How many people have to be recruited to guarantee that 1000 or more will have the same birth sign? Give the minimum number necessary and explain.
(b) If we multiply the number found in a) by 12, is it guaranteeed that we will have 1000 or more people of each birth sign? Support your answer with a brief explanation.

Answer:

Question 1
Applying the addition rule,  alternative GPUs and supported GPUs exist. The game developer will have to test the game on every possible combination of supported CPUs and GPUs. By the product rule, there are  different systems for testing the game.
Question 2
There exist  ways of selecting 2-dog-only owners with each selection having  options of selecting 2-cat only owners. Besides, there are  unique ways of selecting 2-dog and cat owners. By the product rule, the committee can be selected in different ways as follows:
Question 3
The selection of 2 people from each category is ordered implying that the committee could be selected in  ways. That is
Question 4
A minimum of 44 people have to be selected to assure us that we have as a minimum of 4 dog owners and not less 3-cat owners in the sample. We assume that we have 44 people selected. Based on the fact that only 38 people do not own any dog, we have to choose 6 people or more with dogs surpassing the required 4 dogs. After that, we have to select at least 3 cat owners because only 41 people do not have cats. In contrast, a selection of 43 does not assure us that we have at least 4-dog owners and 3-cat owners since we could have 41 dog owners and 2-cat owners in the sample.
Question 5
Part a
Applying the “Generalized Pigeon Hole Principle,” the smallest number of N objects given that  boxes and  is given as . If we select at most 11988, we could select 999 from each group since . In contrast, if we have at most 999 selections from each group, then we could choose 11988 from each birth sign. As a result, selecting not less than 11989 people guarantees 1000 people or more from a birth sign.
Part b
people do not have to be evenly distributed among the 12 boxes denoting that the selection is not guaranteed. Besides, a different distribution could occur, or the entire population sample could have the same birth sign. The study does not guarantee the number of people with a particular birth sign; however, it does guarantee at least a given number of people will have a similar birth sign, but it fails to disclose the type of birth sign.

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