MTH1091 Mathematical Explorations Colorado State University. Global Campus

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MTH109-1 – Mathematical Explorations
Colorado State University – Global Campus
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Introduction
A factorial is a product of a number by itself and the previous numbers and it is represented using an exclamation mark, while variations are used to calculate the permutations. A permutation is a variation where the elements are not repeated, but order of the elements is important. The notation for permutations is P (n, r), which is the number of permutations of "n" elements if only "r" is selected. A variation is also a choice in the arrangement of elements in several different ways. There are different possible ways to consider combination, depending on whether or not there is repetition of the elements or not (Aufmann & Lockwood, 2014). When one repeats the figures there are many more possibilities compared to when there is no repetition. Thus, combinations are ways of grouping the elements, which can be combination without repetition and combination with repetition. This is an analysis using combinations and factorials in three cases where there are eight tomato varieties, unique types of pizza and a bartender concocting drinks using different ingredients.
Tomato varieties
When there are eight tomatoes listed in a seed catalog and five packages of tomato seeds, there are many different ways to choose the seeds from the tomatoes. There is no need to determine how the different elements are ordered, since this does not influence how the packages of tomato seeds represent different tomato varieties (Freund, 2012). In the situation where each of elements take different values and the elements are independent of each other, then calculating the total number of possibilities depends on the combination of values.
8C5= 8! ÷ (5! (8-5)!) = 56
In the combination formula nCr, “n” is the set or population while “r” is a subset of “n” or the sample set. In this case, 8C5 is used to determine the different ways one can choose the tomato seeds from the eight tomato varieties. Using the fundamental counting principle, the number of results in a sample space is the product of the number of results for each element. Furthermore, using a factorial makes it easier to represent permutations and combinations since a factorial is the product of all complete numbers from 1 to a given number. There are 56 ways to select five packages of tomato seeds when a choice represents a different tomato variety.
Types of pizzas Determining the unique types of pizza that can be created is possible when using combination. One can choose from two of seven types of meat, two of eight types of vegetables, one of two types of sauces, and one of three types of crusts. When there are different probability situations with various dependent events, then combining and sorting the results of the different events makes it easier to determine the probability of the events. Since the sample and the size of the event helps to find probabilities, it is useful to know how many combinations are possible. Using the nCr combinations the 7 meats 2 ways (7C2), 8 vegetables 2 ways (8C2), 1 of 2 t...