Permutations, Combinations, and the Binomial Theorem

Transcription

1 Permutatios, ombiatios, ad the Biomial Theorem Sectio Permutatios outig methods are used to determie the umber of members of a specific set as well as outcomes of a evet. There are may differet ways to display possible choices from tables, lists, or tree diagrams ad the cout the umber of outcomes. Oe method of fidig the umber of outcomes is to use the fudametal coutig priciple. This priciple states that if oe task ca be performed ways ad aother task mways. The the two tasks ca be performed mways Example Arragemets with or without restricitios a) How may three digit umbers ca you make usig the digits -, with o repetitio? st umber possibilities all umbers d umber possibilities umbers (ca t use the umber you used first) rd umber possibilities umbers (ca t use the umbers i the first two) Total umber of possibilities is 0 b) How does this chage if repetitio was allowed? st possible umbers d possible umbers rd possible umbers Total Number - ( )( )( ) If you are multiplyig umbers together ca be writte i cosecutive descedig order for example this ca be simplified as!. This is read as five factorial A factorial is for ay positive iteger, the product of all the positive itegers up to ad icludig. Example 0! 0( 9)( 8)( ) or this ca be writte as 0 ( 9)( 8)(! ) If you have 0! it is defied as Permutatio The arragemet of objects or people i a lie is called a liear permutatio. A permutatio is a ordered arragemet or sequece of all or part of a set. For example the possible permutatios of the letters A, B ad are AB, AB, BA, BA, AB, ad BA Oe of the most importat thigs for permutatios is that order matters. The otatio of Pr is used to represet the umber of permutatios or arragemets i a defiite order, or! ritems take from a set of distict items. A formula for r P r, N r! P is If there are seve members o studet coucil, i how may ways ca the coucil select three studets to be the chair, the secretary ad treasurer of the coucil

2 Usig permutatio otatio, Prepresets the umber of arragemets of three objects take from a set of seve objects:! P! 0 So there are 0 ways that the positios ca be filled from the member coucil.!! Example Usig Factorial Notatio a) Evaluate P! P!!! ( 0)( 9)( 8)(! ) ( 0)( 9)( 8) 900! b) Show that!! 9(! )!! (! ) (! )! (( ) )! ( 9) Factor out a! c) Solve for if P!! ( )( )! ( )! ( ) 8 0 ( 8)( + ) 8, 0 Defiitio of Factorial a oly be a positive iteger Permutatios with Repeatig Objects osider the umber of four letter arragemets possible with usig the letters from the work book book obok oobk ookb book obko okbo bkoo kboo kobo koob book obok oobk ookb book obko okbo bkoo kboo kobo koob How do you kow which o is o? If all the letters were differet, the umber of possible four letter arragemets would be! o, which if they were differet, could be arraged i! ways! The umber of four letter arragemets possible whe two of the letters are the same is! There are two idetical letters A set of objects with aof oe kid that are idetical, bof a secod kid that are idetical, ad cof a third kid! that are idetical ad so o ca be arraged i differet ways. a! b! c!...

3 Example Repeatig Objects a) How may differet eleve letter arragemets ca you make usig the letters of mississipp i? There are i s ad s s ad p s! So 0!!! b) I how may differet ways ca you walk from A to B i a three by five rectagular gird if you must oly move dow or to the right? If you thik that you ca oly move right (R) ad dow (D) the you could move DDDRRRRR. So we eed to fid how may differet arragemets that could be. 8! 00 The umber of differet paths is!! 0 Example : Permutatios with ostraits How may ways ca oe frech poster, math posters ad sciece posters be arraged i a row o a wall if a) The two math posters must be together o a ed? b) The three sciece posters must be together? c) The three sciece posters caot all be together? a) Sice the two math posters must be at the ed the would be restricted Poster Poster Poster Poster Poster Poster out of the optios out of optios left optios left optio left out of the math the sciece frech math You would the have to fid the total optios ad the divide by the three sciece posters (!) 8 So total would be 8! b) There are! ways to arrage the three sciece posters so we will cosider this oe object. This meas! that there are objects i total to arrage ways! c) If the three sciece posters ca ot be together the we must look at the total possible umber of arragemets ad subtract the umber of arragemets that have the three sciece posters together!!! To solve some problems you must cout the differet arragmets i cases. For example you might eed to determie the umber of arragemets of four girls ad three boys i a row of seve seats if the eds of the rows must be either both female or both male ase : Girls o Eds of Rows Girl ( girls ad Boys) Girl Arragemets! (! ) 0 ase : Boys o Eds of Rows Boy ( girls ad Boy) Boy Arragemets! ( )(! ) 0 Total umber of arragemets: A B

4 Example Usig ases to determie Permutatios How may digit odd umbers ca you make usig the digits to if the umbers must be less tha 000? No digits are repeated ase Numbers that are odd startig with, or Numbers must start with, or, so there are three choices for the first digit. Numbers are odd, so there are three choices for the last digit ( remaiig ad the ) st Digit d / rd Digit th Digit Arragemets ()() ( )( ) 80 ase Numbers that are odd start with, or st Digit d / rd Digit th Digit Arragemets Total Number of Arragemets: ()() 0 Try these. ) Evaluate the followig expressios a) 8 b) c) d) 9 ) What is the value of each expressio a) 9! b) 9!! (! ) 0! 00 d) (! ) e) (! )(! ) c) (!)(! ) f)!! ) I how may differet ways ca you arrage all the letters of each word a) hoodie b) decided c) aqilluqqaaq d) deeded e) puppy f) baguette ) Solve for the variable a) P 0 b) P 990 c) r 0 P d) ( P ) 0 ) Determie the umber of pathways from A to B a) b) B c) A B B A ) I how may ways ca four girls ad two boys be arraged i a row if a) The boys are o each ed of the row? b) The boys must be together? c) The boys must be together i the middle of the row? A ) How may six letter arragemets ca you make usig all the letters A, B,, D, E, ad F without repetitio? Of these how may begi ad ed with a cosoat? 8) How may itegers from 000 to 8999 iclusive, cotai o s? 9) Usig your uderstadig of factorial otatio ad the symbol Pto r solve each equatio a) P r! b) P r! c) P ( P ) d) ( P ) P

5 Sectio - ombiatios Sometimes i some situatio order is ot importat i the arragemets. For example whe addressig a evelope, it is importat to write the six character postal code i the correct order (permutatio) but addressig a evelope, affixig a stamp ad isertig the cotets ca be completed i ay order. ombiatio Is a selectio of objects of a group of objects take from a larger group for which the kids of objects selected is importat but ot the order i which they are selected. The otatio of ris used to represet the umber of combiatios of items take from a set of rat a time, where r ad r 0. A formula for this is Pr r r!! ( r)! r!! ( r)! r! The umber of ways of choosig three digits from five digits is!!!!!! ( ) 0 There are 0 differet ways to select three items from five Example ombiatios ad fudametal outig Priciple I how may ways ca the debatig club coach select a team from grade studets, ad seve grade studets if the team has a) four members b) four members, oly oe of whom is i grade!!!!!! 9!! ( 0) There is differet ways ( )!! ( )!! 0!!! ( ) There are 0 differet ways!!

6 Example ombiatios with cases A bag cotais seve black balls, ad six red balls. I how may ways ca you draw groups of five balls if at least three must be red? ase Three Red, black!!!!!!!!! ( ) 0 ase All five red!!!!!! ase Four Red, black!!!!!!!!! ( ) 0!!! Total would be Example Simplifyig Expressios ad solvig Equatios with ombiatios a) Express i factorial otatio ad simplify b) Solve for if 0( ) + P! ( + )! 0 ( )!! ( + )! ( )! 0! ( + )! ( )!(! ) ( )!! ( )! ( )! ( + )!( )! ( )!! ( )!! ( )! ( )!(! ) ( )!(! ) ( ( + )(! )( )! )!!! ( )( )! ( )( )! (! ) ( ) ( ) ( )!(! ) ( )! Try these. ) Evaluate a) b) 8 c) d) 0 ) From te employees, i how may ways ca you a) Select a group of b) Assig four differet jobs

7 ) Solve for a) 0 b) c) d) + ) A jury pool cosists of wome ad 8 me a) How may perso juries ca be selected? b) How may juries cotaiig seve wome ad five me ca be selected? c) How may juries cotaiig at least 0 wome ca be selected? ) A pizzeria offers te differet toppigs a) Is this a permutatio of a combiatio questios? Explai. b) How may differet four toppigs pizzas are possible? Sectio - The Biomial Theorem If you expad a power of a biomial expressio, you get a series of terms x + y x + x y + x y + xy + y Are there ay patters here? The coefficiets i a biomial expasio ca be determied from Pascal s triagle. I the expasio of ( x + y), where N, the coefficiets of the terms are idetical to the umbers i the ( + )th row of Pascals triagle. Follow the patter ad fill i the last row. Biomial Pascal s triagle i Biomial Expasio Row ( x + y) 0 x + x + y ( y) ( x + y) ( x + y) ( x + y) + xy x + y + x y + xy x + y + x y + x y + xy x + y The coefficiets i a biomial expasio ca also be determied usig combiatios. Pascal s Triagle ombiatios Patters i Pascal s triagle: Each row begis ad eds with Each umber i the iterior of ay row is the sum of the umbers to its left ad right i the row above Each row is symmetrical Each diagoal has a patter Example Expad Biomials Expad ( c + d ) Look at bottom row of Pascal s triagle for the coef iciets ( c + d ) c + c d + 0c d + 0c d + cd + d How may terms are i this expasio?

8 You ca use the Biomial Theorem to expad ay power of a biomial expressio: 0 x + y x y + x y + x y x y + 0 ( x) ( y) o Example Use the Biomial Theorem Expad ( a b) Substitute a for x ad -b for y i the biomial expasio, so 0 b a b + a b + a b + a b + a 0 b ( a ) o ( a )() + (8a )( b) + (a )(9b ) + (a)( b ) + ()(8 b Try these ) Use the Biomial Theorem to expad: a) a 9a b + a b ab + 8b ( x + y) b) ( p ) c) ) ( a +) ) Expad ad simplify usig the Biomial Theorem: a) ( a + b) b) ( a b) c) ( x ) ) Determie the simplified value of the specified term: a) the sixth term of 9 ( a + b) b) the fourth term of ( x y) c) the seveth term of ( t) ) a) Determie the sum of the umbers i each of the first rows of Pascal s triagle. b) What is a expressio for the sum of the umbers i the ith row of Pascal s triagle. c) What is a formula for the sum of the umbers i the th row? ) Aswer the followig questios for ( x + y) without expadig or computig all its coefficiets. a) How may terms are i the expasio? b) What is the simplified fourth term i the expasio? c) For what value of r does r give the maximum coefficiet? What is that coefficiet? ) a) Expad b) Show that ( x + y) ad ( x y). How are the expasios differet? c) What is the result for ( x + y) + ( x y) x ( x + y ) ( x + y) - ( x y)? How do the aswers i parts (b) ad (c) compare? ) Expad ad simplify usig the Biomial Theorem: a a a) + b) a b b c) x