You should respond to at least two of your peers by extending or adding supplementary distinctions to their posts. Your reply posts are worth 40 points (20 points per response).

Fundamental Counting Principle Problem 

Vera Brito Delgado 

St. Thomas University 

Applied Statistics. STA 2023 AP1 

Dr. Freddy Suarez 

03/30/2023 

Fundamental Counting Principle Problem 

   The fundamental counting principle (FCP) is a mathematical formula frequently used to evaluate the total number of alternative arrangements of a set of objects. These calculations can only solve several probability problems (Yale National Initiative, n.d.). According to the Fundamental Counting Principle, if one event or decision has x possible outcomes or choices and another event has y possible outcomes or options, then the total number of distinct combinations of results between the two is equal to x*y (Nagwa, 2023).  

   For instance, five patients are in critical condition and need ICU care. However, the hospital can only admit three to the ICU at any time due to resource limitations. Using the Fundamental Counting Principle, I will find how many distinct ways the hospital can allocate the five patients among the three admissions allowed.   

   The number of options that are accessible for each phase in the process of allocating patients to the ICU must be determined to solve this problem by utilizing the Fundamental Counting Principle. There are five possible ways to choose a patient for the first admission. Four choices are available for the second admission because a patient has already taken the first admission. There are three options for the third admission since already two patients have been assigned to the first and second admission. Moreover, the total number of possible ways to allocate three admissions among five patients is 5*4*3=60. Furthermore, in this case, using the fundamental counting principle, the hospital can allocate the three admissions to the five patients in sixty different ways. As a result, these calculations can assist the hospital, healthcare professional, or nurse prioritize patient care needs depending on the severity and optimizing resource use. 

References 

Mastin, L. (2020). Fundamental counting principle- Explanation and Examples. Story of mathematics. Retrieved https://www.storyofmathematics.com/fundamental-counting-principle 

Yale National Initiative. (n.d.). 18.04.09: Enumerating daily life with counting principles, permutations, and combinations. https://teachers.yale.edu/curriculum/viewer/initiative_18.04.09_u 

Statistics: Fundamental counting principle

Glenda Garrido Blanco 

St. Thomas University

STA-2023-AP1-Applied Statistics

Dr. Freddy Suarez

March 29, 2023

Fundamental counting principle

Suppose for a moment that you are responsible for choosing the various seating configurations for a restaurant that will shortly be opening. Regarding the interior design of the restaurant, you have two alternatives to pick from. Choices 1 and 2 have a total of two sections, with six tables for each piece. Option 1 is separated into three sections. If you select either Option 1 or Option 2, how many options are there to arrange the tables in the restaurant?

The Fundamental Counting Principle (FCP) is an effective method for determining the total number of potential outcomes in a sequence of events. The rule asserts that if there are m ways to do one task and n ways to complete another, then there are m times n methods to complete both tasks simultaneously. Let's apply this concept to the previously given issue (Wang et al.,2019).

If Option 1 is chosen, there are three unique regions, each containing six tables. Before combining the layouts of the various sections, we must thus organize the tables in each section independently. In the first half of the course, there are six alternative ways to organize the tables. Given that there are six tables and each table may be positioned in one of six ways, there are six possible configurations. ways (Dawadi et al.,2021). Likewise, there are six unique ways to organize the tables in the second and third sections of the presentation! As a direct result of this, there are a total of three alternative ways to organize the tables in Option 1.

6! x 6! x 6! = 466,560,000

Option 2 consists of two divisions, each of which has eight tables. Hence, before we can integrate the two distinct layouts, we must first arrange the tables in each section separately. In the first section of the class, there are eight unique table configurations. Because there are eight tables, there are eight possible combinations! ways. Similar to the first section, there are eight unique methods to structure the tables in the second section. As a direct result, there are three alternative methods to table organization in Option 2.

8! x 8! = 1,032,064,000

Using the fundamental counting concept, we determined that there are a total of 466,560,000 unique ways to arrange the tables in Option 1 and 1,032,064,000 unique ways in Option 2. Hence, if we wish to enhance the number of available seating options in our restaurant, we must select Option 2.

                                                                                           References.

Dawadi, S., Shrestha, S., & Giri, R. A. (2021). Mixed-methods research: A discussion on its types, challenges, and criticisms.  Journal of Practical Studies in Education,  2(2), 25-36.

Wang, S., He, D. Y., Yin, Z. Q., Lu, F. Y., Cui, C. H., Chen, W., … & Han, Z. F. (2019). Beating the fundamental rate-distance limit in a proof-of-principle quantum key distribution system.  Physical Review X,  9(2), 021046.

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