How Combinations and Permutations Differ

Throughout mathematics and statistics, we need to know how to count. This is particularly true for some probability problems. Suppose we are given a total of n distinct objects and want to select r of them. This touches directly on an area of mathematics known as combinatorics, which is the study of counting. Two of the main ways to count these r objects from n elements are called permutations and combinations. These concepts are closely related to one another and easily confused. What is the difference between a combination and permutation? The key idea is that of order. A permutation pays attention to the order that we select our objects. The same set of objects, but taken in a different order will give us different permutations. With a combination, we still select r objects from a total of n, but the order is no longer considered. An Example of Permutations To distinguish between these ideas, we will consider the following example: how many permutations are there of two letters from the set {a,b,c}? Here we list all pairs of elements from the given set, all the while paying attention to the order. There are a total of six permutations. The list of all of these are: ab, ba, bc, cb, ac and ca. Note that as permutations ab and ba are different because in one case a was chosen first, and in the other a was chosen second. An Example of Combinations Now we will answer the following question: how many combinations are there of two letters from the set {a,b,c}? Since we are dealing with combinations, we no longer care about the order. We can solve this problem by looking back at the permutations and then eliminating those that include the same letters. As combinations, ab and ba are regarded as the same. Thus there are only three combinations: ab, ac and bc. Formulas For situations we encounter with larger sets it is too time-consuming to list out all of the possible permutations or combinations and count the end result. Fortunately, there are formulas that give us the number of permutations or combinations of n objects taken r at a time. In these formulas, we use the shorthand notation of n! called n factorial. The factorial simply says to multiply all positive whole numbers less than or equal to n together. So, for instance, 4! 4 x 3 x 2 x 1 24. By definition 0! 1. The number of permutations of n objects taken r at a time is given by the formula: P(n,r) n!/(n - r)! The number of combinations of n objects taken r at a time is given by the formula: C(n,r) n!/[r!(n - r)!] Formulas at Work To see the formulas at work, let’s look at the initial example. The number of permutations of a set of three objects taken two at a time is given by P(3,2) 3!/(3 - 2)! 6/1 6. This matches exactly what we obtained by listing all of the permutations. The number of combinations of a set of three objects taken two at a time is given by: C(3,2) 3!/[2!(3-2)!] 6/2 3. Again, this lines up exactly with what we saw before. The formulas definitely save time when we are asked to find the number of permutations of a larger set. For instance, how many permutations are there of a set of ten objects taken three at a time? It would take awhile to list all the permutations, but with the formulas, we see that there would be: P(10,3) 10!/(10-3)! 10!/7! 10 x 9 x 8 720 permutations. The Main Idea What is the difference between permutations and combinations? The bottom line is that in counting situations that involve an order, permutations should be used. If the order is not important, then combinations should be utilized.

Taking a Look at Autism - 740 Words

At the present moment there are no medications that alleviate the core symptoms of autism. However for children who are aggressive, self harming, hyperactive and engage in repetitive behaviours there are two medications that have been found to be effective; risperidone and aripiprazole. One of these medications could then be prescribed to reduce the engagement in repetitive behaviours since the boy does not exhibit any other symptoms. However this is only for a short period of time until the other therapies can be started. After all, there is a huge risk in taking these medications which is the possible development of diabetes . Applied behaviour analysis Secondly, applied behaviour analysis (ABA) has been proven to be effective in dealing with the core symptoms of autism. It utilizes techniques such as operant conditioning, incidental teaching, shaping for positive reinforcement, discrimination learning, task analysis, among the many to teach the child positive behaviours. Behaviour al interventions such as ABA have found to work best in autism cases. However since behavioural techniques work better when the child is smaller we must use other approaches to compensate for that limitation (Matson et al., 2012; Zachor Itzchak, 2010). Eclectic approach Along with ABA, the eclectic approach could also be used. It integrates philosophies from different approaches to provide a combined method of treatment. In this approach, there are many types ofShow MoreRelatedAutism Spectrum Disorder688 Words   |  3 Pagesterm paper is over Autism. When exposed to the chemicals thalidomide and valfforic acid a person’s risks of getting autism are increased. This is an environmental factor of autism. Scientist know that autism is strongly influenced by genetics. Researchers have found that gene mutations are linked to autism. These genes generally affect synapses. SYNGAP1 is estimated to cause disabilities in about one million people around the world. 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Acrylic Polymer free essay sample

When modern man discovered plastic and its myriad of uses, there was no turning back. Today, plastics figured prominently in the lives of 21st century people. It is almost impossible to live comfortably without this substance. And among the many kinds of polymer there is perhaps nothing more versatile and very much in demand than acrylic polymers. This paper will look into the chemical properties and the myriad of uses for the compound normally designated as PMMA or Polymethyl methacrylate. Chemistry Acrylic acid (H2C – CHCOOH) was synthesized in 1843 according to Charles Carraher (2003, p. 160). Then he goes to add that ethyl methacrylate (H2C – C(CH3)COOC2H5) was synthesized and polymerized in 1865 and 1877, respectively (2003). With regards to polymerization e.g. acrylic or methacrylic acid, it must be done in an aqueous solution at less than 20-30% concentration (Feldman Barbalata, 1996). â€Å"Besides water, a conventional recipe involves an initiating system (potassium persulfate and sodium thiosulfate) and chain transfer agent† (Feldman Barbalata, 1996). Acrylic polymer is derived from the monomer methyl metharcrylate or MMA. Polymers can be divided into two groups: thermoplastics and thermosets. And that majority of them are thermoplastics meaning once the polymer is formed it can be heated and reformed over and over again (see American Chemistry Council, 2007). This major attribute alone is enough to make scientists and businessmen to be very interested in the application of the product. But interestingly there are many more surprises up the sleeves of the thermoplastic polymer. The following attributes can be seen in most polymers and especially so to acrylic polymers: 1.  Ã‚  Ã‚  Ã‚  Ã‚   Has a light transmittancy of about 92% with particular clarity at lower wavelength of 270 to 350 nm (Whitaker, 1996) 2.  Ã‚  Ã‚  Ã‚  Ã‚   Has good resistance to weathering 3.  Ã‚  Ã‚  Ã‚  Ã‚   Good mechanical strength 4.  Ã‚  Ã‚  Ã‚  Ã‚   Dimensional stability along with high tensile and flexural strength (Whitaker, 1996) 5.  Ã‚  Ã‚   Chemical resistance – including salts, bases, aliphatic hydrocarbons, fats and oils, most common gases, and inorganic chemicals as well as dilute and concentrated solutions of most alkali Applications According to Feldman and Barbalatat, the most important usage of acrylic polymer is in the manufacture of adhesives, thickeners, coatings, flocculants, dispersants, fluidizers, in cosmetics, pharmaceuticals, ion exchange processes and textile processing (1996). Now, a few years after acrylic polymer was first introduced by the scientific community, the said substance was first used as interlining for automobile windshields (Carraher, 2003). Since acrylic polymers exhibit good resistance to weathering it is widely used in thermoformed signs, aircraft windshields, and bathtubs. Acrylic polymers have long been used in the manufacturing of cuvettes, tubing connectors, speculums,and many other medical devices requiring impact strength, chemical resistance, biocompatibility, and clarity. In fact it occupies a, â€Å"†¦prominent place in the market for clear, disposable plastic – only glass transmits light as well† (Whitaker, 1996). In the medical field, acrylic polymers have shown its versatility and all around usefulness. From the start medical professionals were already well acquainted with the substance because it was used in the manufacture of incubators. In 1955 its range of usage was expanded when the first acrylic prosthesis was implanted. This probably came after its success as a primary material for aircraft canopies during World War II – pilots suffered fewer infections from shards of acrylic than they had from glass (Whitaker, 1996). Aside as an important component of cuvettes and tubing connectors, it is also used to produce test kits, luers, drainage wands, syringes, blood filters, blood pump housings, fluid silos, surgical blade dispensers, surgical trays etc. Finally, acrylic polymers is not only superior in terms of chemical resistance, clarity as a thermoplastic, and durability which is highly prized in the medical field. This substance is also easily recyclable, â€Å"Acrylic burns extremely clean [†¦] with end products of carbon dioxide and water. In addition, the material offers superior recyclability: acrylic can be reground and reused, which results in less material waste during molding† (Whitaker, 1996). But there is more. The said material can be depolymerized back to its monomer, thoroughly purged of its impurities, and then can go full circle back into PMMA. This is a true recycling process, whereas most other recycling processes involve crushing the material and using it in applications with lower specifications (Whitaker, 1996). In the world of plastic, acrylic polymers belong to the top shelf. The range of its applications is a great help in the development of many industries especially in the medical field. Among other things it has provided cost-efficiency and safety concerning the medical practice and other industrial purposes. References American Chemistry Council. (2007). The Basics – Polymer Definition and Properties. PlasticResource.com [online] Accessed 08 May 2007 from http: www.plasticresource.com/s_plasticresource/sec.asp? TRACKID=CID=124DID=226 Carraher, C. (2003). Giant Molecules: Essential Materials for Everyday Living and Problem Solving. New Jersey: John Wiley Sons, Inc. Feldman, D. Barbalata, A. (1996). Synthetic Polymers: Technology, Properties, Applications. New York: Chapman and Hall. Whitaker, W. (1996). Acrylic Polymers: A Clear Focus. Medical Device Industry [online] Accessed 08 May 2007 from http: www.devicelink.com/mpb/archive/06/01/001.html.

Price Elasticity free essay sample

There are several examples that come to mind when I think of price elasticity. Included in my list are fuel, cigarettes, electricity, and toilet paper. Price elasticity means that the behaviors of supply and demand are not affected when the price of that particular item rises (changes). Our local power companies experience price elasticity on the energy that we demand, when they continually raise prices but the amount of consumer usage is unaffected.In some parts of the country their may be alternatives such as gas heat and fire places which would all contribute to less usage thereby decreasing the likelihood of the price of being inelastic since there would be substitutions. So, rate increases to our power bills increases revenue for the power companies. There is little to analyze since there are no complements. The demand for energy is inelastic so total revenue increases. Elasticity is when there are few variables to change the consumption habits in this particular example, thereby resulting in increased total revenues. We will write a custom essay sample on Price Elasticity or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page However, when there are considerably more variables affecting supply and demand; substitutions is one variable that may actually cause a decrease in total revenue because the price of power has now become elastic when consumers refuse to use power because of the rate increases and they have an alternative source. According to our text and investopedia price elasticity means that supply and demand are affected by changes in the price of an item. A small change in price is accompanied by a large change in demand for the product when the product is elastic.