Solutions for Challenging Counting Problems

Counting can seem like an easy task to perform. As we go deeper into the area of mathematics known as combinatorics, we realize that we come across some large numbers. Since the factorial shows up so often, and a number such as 10! is greater than three million, counting problems can get complicated very quickly if we attempt to list out all of the possibilities. Sometimes when we consider all of the possibilities that our counting problems can take on, its easier to think through the underlying principles of the problem. This strategy can take much less time than trying brute force to list out a number of combinations or permutations. The question How many ways can something be done? is a different question entirely from What are the ways that something can be done? We will see this idea at work in the following set of challenging counting problems. The following set of questions involves the word TRIANGLE. Note that there are a total of eight letters. Let it be understood that the vowels of the word TRIANGLE are AEI, and the consonants of the word TRIANGLE are LGNRT. For a real challenge, before reading further check out a version of these problems without solutions. The Problems How many ways can the letters of the word TRIANGLE be arranged?Solution: Here there are a total of eight choices for the first letter, seven for the second, six for the third, and so on. By the multiplication principle we multiply for a total of 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 8! 40,320 different ways.How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in that exact order)?Solution: The first three letters have been chosen for us, leaving us five letters. After RAN we have five choices for the next letter followed by four, then three, then two then one. By the multiplication principle, there are 5 x 4 x 3 x 2 x 1 5! 120 ways to arrange the letters in a specified way.How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in any order)?Solution: Look at this as two independent tasks: the first arranging the letters RAN, and the second arranging the other five letters. There are 3! 6 wa ys to arrange RAN and 5! Ways to arrange the other five letters. So there are a total of 3! x 5! 720 ways to arrange the letters of TRIANGLE as specified.How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in any order) and the last letter must be a vowel?Solution: Look at this as three tasks: the first arranging the letters RAN, the second choosing one vowel out of I and E, and the third arranging the other four letters. There are 3! 6 ways to arrange RAN, 2 ways to choose a vowel from the remaining letters and 4! Ways to arrange the other four letters. So there are a total of 3! X 2 x 4! 288 ways to arrange the letters of TRIANGLE as specified.How many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN (in any order) and the next three letters must be TRI (in any order)?Solution: Again we have three tasks: the first arranging the letters RAN, the second arranging the letters TRI, and th e third arranging the other two letters. There are 3! 6 ways to arrange RAN, 3! ways to arrange TRI and two ways to arrange the other letters. So there are a total of 3! x 3! X 2 72 ways to arrange the letters of TRIANGLE as indicated.How many different ways can the letters of the word TRIANGLE be arranged if the order and the placement of the vowels IAE cannot be changed?Solution: The three vowels must be kept in the same order. Now there are a total of five consonants to arrange. This can be done in 5! 120 ways.How many different ways can the letters of the word TRIANGLE be arranged if the order of the vowels IAE cannot be changed, though their placement may (IAETRNGL and TRIANGEL are acceptable but EIATRNGL and TRIENGLA are not)?Solution: This is best thought of in two steps. Step one is to choose the places that the vowels go. Here we are picking three places out of eight, and the order that we do this is not important. This is a combination and there are a total of C(8,3) 5 6 ways to perform this step. The remaining five letters may be arranged in 5! 120 ways. This gives a total of 56 x 120 6720 arrangements.How many different ways can the letters of the word TRIANGLE be arranged if the order of the vowels IAE can be changed, though their placement may not?Solution: This is really the same thing as #4 above, but with different letters. We arrange three letters in 3! 6 ways and the other five letters in 5! 120 ways. The total number of ways for this arrangement is 6 x 120 720.How many different ways can six letters of the word TRIANGLE be arranged?Solution: Since we are talking about an arrangement, this is a permutation and there are a total of P( 8, 6) 8!/2! 20,160 ways.How many different ways can six letters of the word TRIANGLE be arranged if there must be an equal number of vowels and consonants?Solution: There is only one way to select the vowels we are going to place. Choosing the consonants can be done in C(5, 3) 10 ways. There are then 6! ways to arrange the six letters. Multiply these numbers together for the result of 7200.How many different ways can six letters of the word TRIANGLE be arranged if there must be at least one consonant?Solution: Every arrangement of six letters satisfies the conditions, so there are P(8, 6) 20,160 ways.How many different ways can six letters of the word TRIANGLE be arranged if the vowels must alternate with consonants?Solution: There are two possibilities, the first letter is a vowel or the first letter is a consonant. If the first letter is a vowel we have three choices, followed by five for a consonant, two for a second vowel, four for a second consonant, one for the last vowel and three for the last consonant. We multiply this to obtain 3 x 5 x 2 x 4 x 1 x 3 360. By symmetry arguments, there are the same number of arrangements that start with a consonant. This gives a total of 720 arrangements.How many different sets of four letters can be formed from the word TRIANGLE?Soluti on: Since we are talking about a set of four letters from a total of eight, the order is not important. We need to calculate the combination C(8, 4) 70.How many different sets of four letters can be formed from the word TRIANGLE that has two vowels and two consonants?Solution: Here we are forming our set in two steps. There are C(3, 2) 3 ways to choose two vowels from a total of 3. There are C(5, 2) 10 ways to choose to consonants from the five available. This gives a total of 3x10 30 sets possible.How many different sets of four letters can be formed from the word TRIANGLE if we want at least one vowel?Solution: This can be calculated as follows: The number of sets of four with one vowel is C(3, 1) x C( 5, 3) 30.The number of sets of four with two vowels is C(3, 2) x C( 5, 2) 30.The number of sets of four with three vowels is C(3, 3) x C( 5, 1) 5. This gives a total of 65 different sets. Alternately we could calculate that there are 70 ways to form a set of any four letters, and subtract the C(5, 4) 5 ways of obtaining a set with no vowels.

To Kill a Mockingbird by Harper Lee Essay - 749 Words

To Kill a Mockingbird by Harper Lee It is not what an author says, but what she whispers which is most important. In other words, one must read in between the lines to discover the subtler meaning of novels. This is true for To Kill a Mockingbird by Harper Lee. Understanding of the many themes in To Kill a Mockingbird is attained only by reading in between the lines. A major one of these themes is dignity as Harper Lee presents a clear picture of which characters are dignified. She does not clearly state which characters are dignified, but she utilizes actions to differentiate between the dignified and the undignified. Amongst the poor folk, the Cunninghams have dignity and the Ewells†¦show more content†¦If one reads To Kill a Mockingbird without pondering the authors title choice, one remains unaware of the books true meaning. Harper Lee characterizes mockingbirds through Atticus warning Jem that it is a sin to kill a mocking bird and Miss Maudie saying that mockingbirds are harmless creatures that only make sweet music for people to enjoy. In their own way, both Boo Radley and Tom Robinson are harmless mockingbirds who do no wrong. As a good neighbor, Boo warms Scout with a blanket and leaves Jem and Scout gifts. Most importa nt, he saves the lives of Jem and Scout by killing Bob Ewell. These achievements are Boos sweet music. Boo is a harmless mockingbird because when he is not doing good deeds, he reclusively stays in his house, never harming people. Harper Lee does not term Boo as a mockingbird, but she endorses this theory through Scout saying that prosecuting Boo would be similar to killing a mockingbird. Furthermore, if Heck Tate had reprimanded Boo, it would have been the equivalent of killing a mockingbird. Tom Robinson is another mockingbird. He lives a peaceful life, minding his own business, except when he charitably helps those he pities. Unfortunately, his benevolence in aiding Mayella Ewell in her chores leads to hisShow MoreRelatedKill A Mockingbird By Harper Lee1049 Words   |  5 PagesTo Kill a Mockingbird: How a Story could be based on True Events in Everyday LifeDaisy GaskinsCoastal Pines Technical Collegeâ€Æ'Harper Lee was born in Monroeville, Alabama. Her father was a former new spaper editor and proprietor, who had served as a state senator and practiced as a lawyer in Monroeville. Also Finch was known as the maiden name of Lee’s mother. With that being said Harper Lee became a writer like her father, but she became a American writer, famous for her race relations novel â€Å"ToRead MoreTo Kill a Mockingbird by Harper Lee1000 Words   |  4 Pagesworld-wide recognition to the many faces of prejudice is an accomplishment of its own. Author Harper Lee has had the honor to accomplish just that through her novel, To Kill a Mockingbird, a moving and inspirational story about a young girl learning the difference between the good and the bad of the world. In the small town of Monroeville, Alabama, Nelle Harper Lee was born on April 28, 1926. Growing up, Harper Lee had three siblings: two sisters and an older brother. She and her siblings grew up modestlyRead MoreKill A Mockingbird By Harper Lee1290 Words   |  6 PagesHarper Lee published To Kill a Mo ckingbird during a rough period in American history, also known as the Civil Rights Movement. This plot dives into the social issues faced by African-Americans in the south, like Tom Robinson. Lee felt that the unfair treatment towards blacks were persistent, not coming to an end any time in the foreseeable future. This dark movement drove her to publish this novel hopeful that it would encourage the society to realize that the harsh racism must stop. Lee effectivelyRead MoreKill A Mockingbird By Harper Lee873 Words   |  4 PagesIn the book, To Kill a Mockingbird, Harper Lee illustrates that â€Å"it’s a sin to kill a mockingbird† throughout the novel by writing innocent characters that have been harmed by evil. Tom Robinson’s persecution is a symbol for the death of a mockingbird. The hunters shooting the bird would in this case be the Maycomb County folk. Lee sets the time in the story in the early 1950s, when the Great Depression was going on and there was poverty everywhere. The mindset of people back then was that blackRead MoreKill A Mockingbird By Harper Lee963 Words   |  4 Pagesgrowing up, when older characters give advice to children or siblings.Growing up is used frequently in the novel To Kill a Mockingbird by Harper Lee. Harper Lee uses the theme growing up in To Kill a Mockingbird to change characters opinion, develop characters through their world, and utilizes prejudice to reveal growing up. One major cause growing up is used in To Kill a Mockingbird is to represent a change of opinion. One part growing up was shown in is through the trial in part two of the novelRead MoreKill A Mockingbird By Harper Lee1052 Words   |  5 PagesTo Kill a Mockingbird by Harper Lee takes place in Maycomb County, Alabama in the late 30s early 40s , after the great depression when poverty and unemployment were widespread throughout the United States. Why is the preconception of racism, discrimination, and antagonism so highly related to some of the characters in this book? People often have a preconceived idea or are biased about one’s decision to live, dress, or talk. Throughout To Kill a Mockingbird, Harper Lee examines the preconceptionRead MoreHarper Lee and to Kill a Mockingbird931 Words   |  4 PagesHarper Lee and her Works Harper Lee knew first hand about the life in the south in the 1930s. She was born in Monroeville, Alabama in 1926 (Castleman 2). Harper Lee was described by one of her friends as Queen of the Tomboys (Castleman 3). Scout Finch, the main character of Lees Novel, To Kill a Mockinbird, was also a tomboy. Many aspects of To Kill a Mockingbird are autobiographical (Castleman 3). Harper Lees parents were Amasa Coleman Lee and Frances Finch Lee. She was the youngestRead MoreKill A Mockingbird By Harper Lee1695 Words   |  7 PagesIn To Kill a Mockingbird Harper Lee presents as a ‘tired old town’ where the inhabitants have ‘nowhere to go’ it is set in the 1930s when prejudices and racism were at a peak. Lee uses Maycomb town to highlight prejudices, racism, poverty and social inequality. In chapter 2 Lee presents the town of Maycomb to be poverty stricken, emphasised through the characterisation of Walter Cunningham. When it is discovered he has no lunch on the first day of school, Scout tries to explain the situation to MissRead MoreKill A Mockingbird By Harper Lee1876 Words   |  8 PagesThough Harper Lee only published two novels, her accomplishments are abundant. Throughout her career Lee claimed: the Presidential Medal of Freedom, Pulitzer Prize for Fiction, Goodreads Choice Awards Best Fiction, and Quill Award for Audio Book. Lee was also inducted into the American Academy of Arts and Letters. This honor society is a huge accomplishment and is considered the highest recognition for artistic talent and accomplishment in the United States. Along with these accomplishments, herRead MoreKill A Mockingbird, By Harper Lee1197 Words   |  5 Pagessuch as crops, houses, and land, and money was awfully limited. These conflicts construct Harper Lee’s novel, To Kill a Mocking Bird. In To Kill a Mocking Bird, Lee establishes the concurrence of good and evil, meaning whether people are naturally good or naturally evil. Lee uses symbolism, characterization, and plot to portray the instinctive of good and evil. To Kill a Mocking Bird, a novel by Harper Lee takes place during the 1930s in the Southern United States. The protagonist, Scout Finch,

Representation makes dummies of us all Essay Example For Students

Representation makes dummies of us all Essay He is still romanticised, to some extent, through his own description of his actions; I dont talk much. I swing up beside them and do it with my eyes. Brando. By alluding to such a well known actor, the reader again questions their own opinion on the speaker how can this Psychopath be just that, but also comparable to Marlon Brando, Jimmy Dean and a king? The identity of the character comes through a construct of others, none of his own traits are apparent, only a mixture of various other identities. Although the character describes himself as crystal, the image is blurred by the breath on the mirror. Mirrors and reflections play a large part in Carol Ann Duffys poetry, especially in Psychopath. The contradiction of crystal clear and the steamed mirror gives the impression of a lack of reflection you cannot see yourself when the mirror is misted up. Without a reflection, then, can there really be a character in the first place? He appears to be clear, yet unclear, real yet unreal even the words he speaks are given to him by the poet. At the end of the poem however, the speaker does identify with his reflection, yet does not refer to it as himself; Heres looking at you. These words, again not his own, are from the film Casablanca in which Humphrey Bogart plays a bad character who is redeemed. Here, however, the character is not redeemed and yet again the speaker is comparing himself to something, rather than being an individual. This is apparent through many of Duffys poems; simile and metaphor are often used as description, comparing to a similar thing, not describing exactly or definitively. Therefore, we are only ever given a list of what something is like, rather than what it actually is. These comparisons are of course subjective, so what we gain is an opinion, a description through vague ideas of similarity. What we are left with then, is just a silhouette, an outline, a dummy of what is really being described. For the character, is appears almost a sense of delusion, as Gregson argued; These tactics allow to suggest how overpoweringly right and reasonable sexist attitudes can appear to those who hold them, and how wrong and aggressive their consequences are for their victims (Gregson 1996 : 106-7) In a sense, it could be seen that using a series of similes, comparisons and also stereotypes, the characters of Duffys poems are hiding their identity, perhaps from themselves; it is easier to say I am like him and like her than accept, or even understand, your own personal identity. Poet for our Times for example, shows a series of truly British caricatures Eastenders sex stories, MPs misbehaving and page three models which satirises the way that British culture obsessively caricatures itself in its concern not to take itself too seriously (Gregson 1996 : 104). The same can be seen in Model Village (Selected Poems : 37) where everything is just how it should be; Grass is green / and the pillar-box is red. The speaker then adds, Wouldnt it be strange if grass were red? Hard to Say (The Other Country : 45 ) looks at tired clichi s used in reference to love, but this is also applicable to the hiding of identity in Duffys poems. It is as if this could be any other couple, in any other time, any other place. The words become plain, the same and meaningless. Words, Wide Night (The Other Country : 47) is another poem about love, which muses on how words cannot represent the feeling between two people. Does then, trying to represent love, take away the power of it, lessen it and make it mundane; a dummy of what it really is? Or rather, use the lazy words of Hard to Say to become just the same as everybody else. Unique identity seems undesirable and Carol Ann Duffy seems to want to highly that the quest for acceptance places a large part in peoples everyday lives this can be seen from the minority voices that Duffy tries to represent. .u460492d94815ec8e4d823c8f44376497 , .u460492d94815ec8e4d823c8f44376497 .postImageUrl , .u460492d94815ec8e4d823c8f44376497 .centered-text-area { min-height: 80px; position: relative; } .u460492d94815ec8e4d823c8f44376497 , .u460492d94815ec8e4d823c8f44376497:hover , .u460492d94815ec8e4d823c8f44376497:visited , .u460492d94815ec8e4d823c8f44376497:active { border:0!important; } .u460492d94815ec8e4d823c8f44376497 .clearfix:after { content: ""; display: table; clear: both; } .u460492d94815ec8e4d823c8f44376497 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u460492d94815ec8e4d823c8f44376497:active , .u460492d94815ec8e4d823c8f44376497:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u460492d94815ec8e4d823c8f44376497 .centered-text-area { width: 100%; position: relative ; } .u460492d94815ec8e4d823c8f44376497 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u460492d94815ec8e4d823c8f44376497 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u460492d94815ec8e4d823c8f44376497 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u460492d94815ec8e4d823c8f44376497:hover .ctaButton { background-color: #34495E!important; } .u460492d94815ec8e4d823c8f44376497 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u460492d94815ec8e4d823c8f44376497 .u460492d94815ec8e4d823c8f44376497-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u460492d94815ec8e4d823c8f44376497:after { content: ""; display: block; clear: both; } READ: Daisy Miller and Catherine Sloper EssayBy subverting identity through a series of other images and stereotypes, it is easy to blend in, become a dummy, just as the ones that the Psychopath sees through the shop windows (Selected Poems : 43). Ironic perhaps, that the good-looking girl makes the air sing Johnny, Remember Me, as she will not remember anything more after her encounter with the Psychopath and neither will he, as he adds, Tomorrow / will find me elsewhere, with a loss of memory. It seems therefore, that we have come full circle; erasing memory will take away identity, leaving a blank, a dummy, which the poet cannot represent. In conclusion, it must be remembered that representation is just that an attempt to represent a concept, an idea or a character, but not the actual thing itself. Every person will interpret that notion or persona in a different light and so putting this into words will also therefore be subjective. This is more extreme in the case of Carol Ann Duffy, due to her radical choice of subject matter and the people she chooses to represent in her poetry. In essence then, of course representation creates dummies, models or imitations of the reality when seen by a wide readership. To Duffy, these representations will be more accurate they are her perceptions of the world and of the characters she has created. However, the reader must accept these perceptions and opinion as, after all, this is what is written; there is no second opinion. Ultimately, the acceptance of these words makes us dummies, with no choice but to accept, attempt to interpret and try to understand, and enjoy Carol Ann Duffys poetry. Word Count : 2013 Primary Texts Duffy, Carol Ann. , 1985. Standing Female Nude. Anvil Press : London Duffy, Carol Ann. , 1987. Selling Manhattan. Anvil Press : London Duffy, Carol Ann. , 1990. The Other Country. Anvil Press : London. Duffy, Carol Ann. , 1993. Mean Time. Anvil Press : London Duffy, Carol Ann. , 1994. Selected Poems. Penguin : London. Duffy, Carol Ann. , 1999. The Worlds Wife. Picador : London Secondary Texts Bakhtin, M. M. , 1981. Holquist, M. ed. , Emerson, C. , trans. The Dialogic Imagination. University of Texas Press : Austin Connell, P. E. , 2005. Raising the subject : Indeterminacy in the poetry of Carol Ann Duffy. University of Hull : Kingston-Upon-Hull Freud, S. , 1948 (1901). The Psychopathology of Everyday Life. Ernst Benn : London Gregson, I. , 1996. Comtemporary Poetry and Postmodernism. Macmillan : Basingstoke Kenigan, J. , 2004. Notes from the Home Front : Contemporary British Poetry. Essays In Critcism. (54,2) Michelis Rowland. , 2005. The Poetry of Carol Ann Duffy : Choosing tough words. Years Work of English Studies, (84,1) pp. 764-853 Porter, E. , 1999. What like is it? Landscape and Language in Carol Ann Duffys Love Poetry. Neohelicon, (26, 1) pp. 79-80 Robinson, A. , 1988. Instabilities in Contemporary British Poetry. Macmillan : Basingstoke Thomas, J. E. , 1998-9. The intolerable wrestle with words: Carol Ann Duffy. Bete Noire, (6) pp. 78-88 1 Alfred Hitchcock Source unknown.

Renaissance Essays (887 words) - Medieval Philosophy, Renaissance

Renaissance Renaissance Essay The 14th, 15th and part of 16th century was a glorious time for Europe, it was the reformation of many old ideas and the formation of many new, this was called the Renaissance. The Renaissance brought many changes to Europe, the economy was greatly boosted by of all the new explorations. The flourishing economy helped to inspire new developments in art and literature. And from that many new beliefs were formed. The European economy flourished during the Renaissance due to many factors. There was a large income coming in from over seas exploration. Spain alone received added income from Christopher Columbus and when he stumbled across North America on his way to find a shorter rout to the Indies. The exploration down the coast of Africa also brought in a lot of extra income. This income came from the exploitation of the Africans by kidnapping them and selling them as slaves. Income also came from establishing colonies in Africa and setting up gold mines and mines for other needed metals. With all of this extra income coming into the European countries they had more money to help fund the arts. During the Renaissance there were many drastic changes in the style of art. Giotto was a very influential painter, during the start of the Renaissance. In Giotto's work he used three dimensional images, this was a drastic change from the classic art where depth was not used. His painting were very realistic and life like, unlike the previous centuries art. Giotto's work inspired artists by the likes of Leonardo de Vinci, Raphael and Michaelango. Michaelango painted the Sistine Chapel, which is located in the Vatican in Rome. The Sistine Chapel depicted the book of Genesis. The manner in which it was painted was unlike another at the time, all the characters in the Sistine chapel are very life like and realistic. Also it was one of the first times that religion was painted by the painters opinion of the events. The arts led to new ways of thinking. With the arts the artists began to think on their own and those movements began to spread. It was not just what the church said anymore that was right. Humanism, one of the new beliefs which was formed during the Renaissance, said that people should read the works of the greats and focus on writing, and the arts. Humanists believed that they were equal with the ancient Greek and Roman writers and philosophers. Petrarch was the original humanist, and a writer who wrote many letters to the people of ancient Rome. In those letters he spoke as if he was an equal with them. Another of the new beliefs was scholasticism, which was the opposite of humanism. Scholastics thought that people should spend more time the sciences, they also wanted the church and science to be brought together as one. As new scientific discoveries were made many of the churches theories were beginning to be questioned. Some of the new scientific discoveries consisted of theories which went against the churches beliefs. One theory which was proven true about the Earth revolving around the sun. This was contrary to the churches view that everything revolved around the Earth. This theory was proven true by Copernicus who is considered the father of modern astronomy. Kelper an astronomer and astrologer also discovered about the rotation of the planets. As the church began to be questioned more and more. And soon there were new religions forming. The major religion that was formed during the Renaissance was the Protestant religion. The Protestant religion began to spread throughout Europe, at one point of time the official church of England was a Protestant church. This new and improved Europe was mainly came to be from the blustering economy, the explorations, art, literature, and new discoveries, but this new growth of a western power was not just a natural development, in fact it was inspired by a superior civilization, the Chinese. If it was not for the Chinese version of the Renaissance which occurred about a thousand years earlier there would be no European Renaissance. The Chinese had many useful inventions which the Europeans exploited unlike the Chinese. One crucial Chinese invention was the