number of distinguishable arrangements of each of the following
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(b)&&&&baaaaaben
(c)&&&&aaaaabbba
Show transcribed image text Find the number of distinguishable arrangements of each of the following "words."
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Permutations With your group find as many arrangements of the letters A, H, M, T as you can. How many 2 letter arrangements are there? Could you do this.
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Presentation on theme: "Permutations With your group find as many arrangements of the letters A, H, M, T as you can. How many 2 letter arrangements are there? Could you do this."— Presentation transcript:
Permutations With your group find as many arrangements of the letters A, H, M, T as you can. How many 2 letter arrangements are there? Could you do this an easier way? Use a tree diagram, or…
Key Terms: Permutation: is any arrangement of items or events Notation: P(n,r) or nPr – n tells you what number to start with, r tells how many numbers you are multiplying Factorial: the product of a whole number and every positive number less than itself. Notation- x! – 0! = 1
Examples: How many different ways can the letters of each word be arranged? 1. SAND 2. GREEN 3. CAT 4! = 4 ● 3 ● 2 ● 1 = 24 5! = 5 ● 4 ● 3 ● 2 ● 1 = 120 3! = 3 ● 2 ● 1 = 6
Examples: Find the value. 4. 7! 5. P(8,2) 6. P(9, 3) 7 ● 6 ● 5 ● 4 ● 3 ● 2 ● 1 = 5040 8 ● 7 = 56 9 ● 8 ● 7 = 504
Examples: 7. In how many ways can six people line up for a photograph? 8. A building inspector is supposed to inspect 10 building for safety code violations. In how many different orders can the inspector visit the buildings? 6! =6●5●4●3●2●16●5●4●3●2●1 720 ways 10 ! =10 ● 9 ● 8 ● 7 ● 6 ● 5 ● 4 ● 3 ● 2 ● 1 = 3,628,800 ways
Examples: 9. How many 3 letter words can you make from 5 letters? 10. How many 4-letter, two digit license plate numbers can you make? P(5, 3) 5●4●35●4●3 60 words 26 ● 26 ● 26 ● 26 ● 10 ● 10 a. If repeat letters and numbers allowed b. If repeat letters and numbers not allowed 26 ● 25 ● 24 ● 23 ● 10 ● 9 45,697,600 plates 32,292,000 plates
Ads by GoogleHow do you find the number of distinguishable arrangements for the letters of the word: GOOGOL?
Explain the theory you used also.
Bonus points if the fundamental counting principle is used.
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In general, the number of distinguishable arrangements for a six letter word is 6! = 720. But that's where all letters are themselves distinguishable.
Here you've got a case where there are 3 indistinguishable elements, since there are 3 O's. So to take this into account, you must divide by 3!. And there are two G's also, so you must further divide by 2! to get
6!/(2!3!) = 720/12 = 60.
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How do you find the number of distinguishable arrangements for the letters of the word: GOOGOL?
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15 answersHow to Calculate the Number of Isomers | eHow
Isomers are compounds that are identical in formula but different in structure or spatial arrangement. They occur throughout nature but are of special interest in organic chemistry -- the study of carbon compounds -- because of the huge variety of economically important organic molecules. Scientists have tried to mathematically derive the number of isomers of straight-chain organic molecules, called alkanes, but they have discovered no simple relationships between isomer count and carbon content. However, computer programs that decompose alkane structures into manageable fragments give good results.
The two types of isomers are the structural and the optical. Structural isomers have different arrangements of atoms or small clusters of atoms, called functional groups. These isomers result from differences in the way the molecules branch and how the functional groups are arranged. Optical isomers, or stereoisomers, are structurally identical but differ in how the atoms and functional groups are geometrically positioned in space. Examples of optical isomers include mirror images and molecules that twist in opposite directions.
Alkanes are chains of carbon (C) and hydrogen (H) atoms, arranged so that for every n carbon atoms there are 2n + 2 hydrogens. Alkanes originate principally from natural gas and crude oil. The carbon in alkanes forms chains in which each carbon binds to four other atoms through either C-C or C-H bonds. Straight, or acyclic alkanes, don't form ring structures. The simplest alkane is methane, CH4. Alkanes with four or more carbon atoms can form structural isomers, and those with seven or more carbons can also form optical isomers. Some isomers are &sterically unfavorable,& meaning that they are unlikely to form because they require extra energy to remain stable.
Robert Paton and Jonathan Goodman at the University of Cambridge offer a free application, called IsoCount, that computes the number of structural and optical isomers for any acyclic alkane. You simply enter the number of carbons in the alkane and the program figures the structural and optical isomer count, noting how many are sterically unfavorable. The program uses an algorithm that iteratively examines portion of the alkane to derive the number of isomers. For example, if you enter seven, the program reports that the C7H16 alkane has nine structural isomers and two optical ones.
Alkanes with 16 or 17 carbons are not stable compounds and would rapidly dissociate at room temperatures. C17 doesn't exist at all and C16 can only form briefly at very low temperatures. Some longer-chain alkanes are also unstable. The IsoCount program accounts for unstable carbon fragments when reporting its results. The count of isomers grows rapidly as the number of carbons in the alkane increases. The IsoCount authors estimate that the isomers of an alkane with 167 carbons would outnumber all of the particles in the universe.
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The convertible senior notes and Solarfun’s ordinary shares represented
by the ADSs, if any, issuable upon conversion of the notes have not been
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resale of the notes and Solarfun’s ordinary shares represented by the
ADSs, if any, issuable upon conversion of the notes and use its
commercially reasonable efforts to cause such registration statement to
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Morgan Stanley & Co. Incorporated is acting as the sole bookrunning
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Concurrently with this offering of notes, Solarfun is offering, in a
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ordinary shares considered outstanding for the purpose of calculating
its basic or diluted earnings per share under U.S. GAAP.
This press release does not constitute an offer to sell or the
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offer, solicitation or sale in any jurisdiction in which such offer,
solicitation or sale would be unlawful. Any offers of the notes will be
made only by means of a private offering memorandum.
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