Binomial Coefficients and Subsets

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1 page Fiite Mathematics Biomial Coefficiets ad Subsets I the worksheet Combiatios ad Their Sums, we saw how ib Mu im, livig i Morocco aroud, used combiatios C (, r) to cout the umber of ways to select ed threads or other thigs from a pool of items, assumig that repeats are t allowed ad the order of selectio does t matter. Let s do some practice ad warm-up. Exercise. a) I how may differet ways ca employees be selected from a office of 7 employees? C (, ) = b) Use the row-sum patter to simplify: C (,) + C(,) + C(,) + C(6,) = C(, ) = c) Use the colum-sum patter to simplify: C ( 9,) + C(9,6) = C(, ) = Ib Mu im was t the first to explore combiatios like these, but apparetly they d ever bee studied by mathematicias i aciet Greece ad Mesopotamia, who excelled i may other topics. I the late 9s, a Idia poet amed Halayudha preseted the arithmetical triagle as a tool i categorizig the differet kids of poetic meter, the sequeces of stressed or ustressed syllables i oe lie of poetry (see Exercise 9 below). A little later, aroud, the Arab mathematicia Abū Bakr al-karajī i Baghdad used the arithmetical triagle to solve a algebra problem, that of figurig out the coefficiets of biomial powers like (see Exercises - below). Durig the Middle Ages, it was the Arab world that was pre-emiet i sciece ad mathematics. I fact, our word algebra came from the title of a importat Arabic text, Hisāb aljabr wa-l-muqābala, writte by the mathematicia al-khowārizmī i Baghdad aroud 8. Scholars such as al-khowārizmī ad al-karajī were associated with libraries, referred to i Arabic at the time as houses of wisdom. These were essetially govermet-supported research ceters, the oe i Baghdad beig the most famous. Scholars speakig may differet laguages arrived there from throughout the Middle East, but they coversed with each other i Arabic (that s why may of our mathematical terms, such as algebra, algorithm ad zero, came to us from Arabic). Al-Khowārizmī Drawig from Mohammad Tūfīq Haydār, Tārīkh al- ulūm ida-l- arabī (Beirut: Dār al-tashi at al-lubāīyyat, 99) Loci: Covergece (August ), DOI:.69/loci6

2 page Exercise. Let s start with oe of the simplest biomial powers: = ( x + y) (use distributivity) = _ + _ + _ + _ F O I L = _ + _ + _ use commutativity to combie the like terms = _ x + _ xy + _ y (fill these blaks with the umerical coefficiets) Sice is a biomial (two-term) expressio, we call a biomial power. We say that we have expaded the biomial power, ad the umerical coefficiets (,, ad ) are called biomial coefficiets. Exercise. Determie the biomial coefficiets of = = ( x + xy + y ) i the same way: = _ + _ + + _ + _ + _ = _ x + x y + xy + _ y To expad a biomial power like, we could keep goig like this, but it would take a lot of time ad effort to work our way up to! To simplify thigs, we look for patters. Let s summarize what we kow from the first four powers of : = = x + y = x + xy + y = x + x y + xy + y Look at the triagular arragemet of coefficiets above. Compare this to ib Mu im s arithmetical triagle, show at right. Do you see the similarity? It seems as if each set of biomial coefficiets is exactly the same as oe colum of the arithmetical triagle, the triagle whose etries were obtaied by calculatig combiatios C (, r). For a tassel of s For a tassel of 9 s 9 For a tassel of 8 s 8 6 For a tassel of 7 s For a tassel of 6 s For a tassel of s 7 6 For a tassel of s 6 8 For a tassel of s For a tassel of s For a tassel of # # (& # (& # (& # (& #6 (& #7 (& #8 (& #9 (& # (& Total o. of tassels Loci: Covergece (August ), DOI:.69/loci6

3 page This suggests that we ca predict biomial coefficiets simply by our combiatio formula C (, r). But which ad r do we plug i? Remember that alog each colum of ib Mu im s triagle, the umber of s available,, was a costat. So, we ll write each coefficiet to have the same first umber, such as : ex. = x + x y + xy + y (,) x + C(,) x y + C(,) xy C(,) y = C + The secod umber, r, which varies from term to term, tells you exactly how may of the variables i the term are y. For example, ext to xy we write C (,), sice of its variables are y. It makes sese that this is the correct way to predict the coefficiet, because the xy term combies the like terms xyy, yxy ad yyx. Sice the umber of such terms is the umber of ways to fill ay of the spots with y (ad the rest of them with x), the correct coefficiet must be C(,) which is. From ow o, we ll write C (, r) i the more compact biomial coefficiet otatio,. r Although calculatig the biomial coefficiet ivolves dividig, you do t write the fractio bar r there, because that would make it look like, whereas the formula for ivolves a lot more tha r r just dividig by r. As a review, r = Cr =! r!( r)! The fastest way to write it by had The way to calculate it o a TI calculator The way to calculate it from scratch Look how much more compact the expasio becomes i this ew otatio: ( x + y + y) = C(,) x + C(,) x y + C(,) xy C(,) (old otatio) ( x y + y) = x + x y + xy + (ew otatio) If you stare at this for a while, you otice patters that ca be used to shortcut the harder problems such as : a) Each term i the expasio has the same total umber of variables,. This is called the total degree of the term. b) Each biomial coefficiet uses the same umber o top,. c) The umber o the bottom, r, icreases systematically from to. d) The bottom umber, r, tells you exactly how may of the variables i the term are y. e) The fial coefficiets are symmetrical: they read the same forwards as backwards, e.g.,,,,. Loci: Covergece (August ), DOI:.69/loci6

4 page These patters ca be summarized i the form of a formula, ofte called the Biomial Theorem: x = x y + x y + x y + + x y + Exercise. Use these patters to complete the followig expasios (fill i all blaks): y a) = x y + x y + x y + x y + x y = x + x y + x y + xy + y (Simplify your aswer.) b) = _ + _ + _ + _ + _ + _ = (Simplify your aswer.) 7 c) = _ + _ + _ + (Fid the first terms oly.) = _ + _ + _ + (Simplify your aswer where possible.) 9 d) ( r + s) = + _ + _ + _ + _ (Fid the last terms oly.) = + _ + _ + _ + _ (Simplify where possible.) Check your work i parts (a) (d) by referrig to the colums of ib Mu im s arithmetical triagle o page. e) ( +.) = _ + _ + _ + _ + (Fid first terms oly.) = (Simplify where possible.) = (Add it up.) f) ( + ) = _ + _ + _ + _ + _ + _ = (Simplify the powers oly.) = (Now check your aswers.) g) CHALLENGE PROBLEM I the expasio of ( r s ) (Show all of your work.) 6, the fully simplified coefficiet of r 9 s 6 is. Loci: Covergece (August ), DOI:.69/loci6

5 page Exercise e suggests that biomial expasios might be useful i umerically approximatig fifth powers ad fifth roots (ad other powers ad roots). This is exactly oe of the ways they were used i medieval Chia. Exercise. Show at the right is the arithmetical triagle give i the algebra textbook Precious Mirror of the Four Elemets (Siyua Yuchia), writte durig the Yua dyasty i by Chu Shi-Chieh. His text focused o solvig systems of may equatios ad variables; the four elemets of Chu s title referred to ma, matter, heave ad earth, which symbolized four variables. a) Comparig this triagle to those we ve already worked with, traslate the followig umerals from Ido-Arabic ito Chiese: + = 6 + = (fill i each circle) b) Rememberig that biomial coefficiets are always symmetrical, see if you ca discover a slight mistake that Chu (or his scribe) made i copyig out his triagle. Record your discovery as the followig violatio of symmetry: (fill i each circle) Photo: Joseph Needham, Sciece ad Civilizatio i Chia (Cambridge Uiversity Press, 99) Triagles like these were also put to use later i Europe. Blaise Pascal (i Frace i 6) used the triagle to solve a probability problem arisig from a gamblig situatio, ad Isaac Newto (i Eglad i 66) used it i cojuctio with his developmet of calculus. I Europe ad the Americas, the triagle is still geerally kow as Pascal s Triagle. Pascal s Arithmetical Triagle from his Traité du Triagle Arithmétique (66) Photo: Cambridge Uiversity Library Loci: Covergece (August ), DOI:.69/loci6

6 page 6 Loci: Covergece (August ), DOI:.69/loci6 Exercise f suggests that i ay biomial expasio, the sum of the biomial coefficiets is a power of. Let s cofirm this. Exercise 6. Show below is a arithmetical triagle i the Chiese style. row o. row total 6 a) Fill i the missig row totals. b) Is each row total a power of? _ c) Based o your observatio, it appears that if stads for the row o., the the row total is = (fill i the missig formula) Exercise 7. A telephoe compay offers customer optios i additio to its basic service. A customer ca choose ay (or all or oe) of the optios. I how may differet ways ca this be doe? Let s solve this by two differet methods. a) (Additio Rule) Let s tally the subsets accordig to the size of the package purchased: The umber of ways to select of the optios is =. The umber of ways to select of the optios is =. The umber of ways to select of the optios is =. The umber of ways to select of the optios is =. The umber of ways to select of the optios is =. The umber of ways to select of the optios is =. The umber of ways to select some subset of the optios is (total).

7 page 7 b) (Multiplicatio Rule) Let s tally the subsets accordig to the customer s decisio process: Step : decide whether to buy optio or ot; = choices. Step : decide whether to buy optio or ot; = choices. Step : decide whether to buy optio or ot; = choices. Step : decide whether to buy optio or ot; = choices. Step : decide whether to buy optio or ot; = choices. Total umber of ways to make decisio =. Comparig your aswers to the two parts of Exercise 7, the fact that they match cofirms why =, or more geerally, =. Wheever you add up all the biomial coefficiets havig the same upper umber (umber of optios available), their total is raised to that power. Iterpretig this aother way, if a set has elemets, the umber of subsets of it (icludig the whole set ad the empty set) is. This rule gives us a powerful shortcut for solvig lots of coutig problems. ex. Suppose that i Exercise 7, the customer is t allowed to choose more tha of the optios. I how may differet ways ca this be doe? = optios available without repeats r =,,, or optios chose without order Log method: = =, as Shortcut: = = =, as Use this shortcut to solve the rest of the exercises. Exercise 8. About CE, the Idia doctor Sushruta used combiatios to compute the umber of flavors composed of oe or more of the 6 basic tastes (bitter, sour, salty, astriget, sweet, hot). Predict the aswer that he foud. = 6 tastes available r =,,,, or 6 tastes selected umber of combiatios = 6 = 6 = _ 6 without repeats without order Loci: Covergece (August ), DOI:.69/loci6

8 page 8 Exercise 9. The great astroomer ad mathematicia of south Idia, Bhāskara (-8), illustrated the use of biomial coefficiets by posig various problems from art ad architecture i his classic work Līlāvatī (traslated by H. T. Colebrooke, Allahabad, Idia: Kitab Mahal, 967). a) I classical Idia poetry ad music, each lie is a combiatio of log (L) ad/or short (S) syllables. To systematically test out the soud quality of differet meters, it was useful to kow the umber of variatios possible. Bhāskara explaied that i a 6-syllable lie, the umber of combiatios havig log syllables (such as LLSSLL) is 6 =. Based o such methods, he asked for the umber of variatios possible (p. 6). = 6 syllables available i a lie r =,,,,, or 6 log syllables umber of combiatios = b) At oe poit, Bhāskara described a pleasat, spacious ad elegat edifice, with eight doors, costructed by a skilful architect, as a palace for the lord of the lad (p. 6). He the asked for the total umber of combiatios of palace doors that could be opeed, assumig that at least oe is opeed. = _ doors available r = _ doors opeed umber of combiatios = Whe we choose which of the 6 syllables will be log, the order of choice does t matter ad repeats are t allowed. Taj Mahal photo: Richard IJzermas, Exercise. I 9, the mathematicia Abū-l- Abbās Ahmad ib al-baā of Marrakech, Morocco, computed the umber of various types of geometry problems. For example, he said that every triagle has basic quatities (height, area, ad the legths of the sides). I ay give triagle problem, betwee ad of these quatities are kow, ad the problem is to try to determie the rest. So, the umber of possible types of triagle problem is: = total quatities r =,, or kow quatities umber of problems = without repeats without order = =, as a) Ib al-baā said that every circle has basic quatities (diameter, perimeter ad area). How may differet types of circle problem are possible? b) Ib al-baā said that every rectagle has basic quatities (legth, width, diagoal ad area). How may differet types of rectagle problem are possible? Loci: Covergece (August ), DOI:.69/loci6

9 page 9 Exercise. A ew kid of combiatio lock has te pushbuttos (labeled through 9). Each butto ca be toggled betwee its up ad dow positio. The lock opes oly if a certai combiatio of the buttos (the subset programmed by the ower) is pressed dow, i ay order. a) Suppose the subset programmed by the ower must iclude at least oe butto i the dow positio. I how may differet ways ca such a lock be programmed? b) Compare this to a covetioal combiatio lock. Suppose that such a lock ca be opeed oly by the correct sequece (the order is importat) of three digits, each digit betwee ad 9. I how may differet ways ca such a lock be programmed? Exercise. I the dim sum restaurat Heavely Garde, a trolley comes aroud with a selectio of 7 dishes. I how may differet ways ca a customer choose as may as of the dishes? Exercise. Pratima is coductig a study examiig the major difficulties faced by miority-owed busiesses i Oaklad Couty. As part of her study, she has idetified 9 such busiesses that declared bakruptcy i the last 8 moths. If she wats to select a sample of betwee ad 8 of these as focuses of case studies, how may differet samples are possible? Loci: Covergece (August ), DOI:.69/loci6