iff s = r and y i = x i for i = 1,, r.

Transcription

1 Combiatoics Coutig A Oveview Itoductoy Example What to Cout Lists Pemutatios Combiatios. The Basic Piciple Coutig Fomulas The Biomial Theoem. Example As I was goig to St. Ives I met a ma with seve wives Evey wife had seve sacks Evey sack had seve cats Evey cat had seve kits Kits, cats, sacks, wives How may wee goig to St. Ives? Patitios Solutios Lists Ca be Couted Easily As: Noe. How may wee goig the othe ways? 7 Wives sacks cats kits. Total Ode Pais: (x, y) (w, z) iff w x ad z y. Odeed Tiples: (x, y, z) (u, v, w) iff u x, v y, ad w z. Lists of Legth (AKA Ode -tuples): (x 1,, x ) (y 1,, y s ) iff s ad y i x i fo i 1,,. Example: Licese Plates. A licese plate has the fom LMNxyz, whee L, M, N {A, B,, Z}, x, y, z {0, 1,, 9} ad, so, is a list of legth six.

2 Basic Piciple of Combiatoics The Multiplicatio Piciple Fo Two: If thee ae m choices fo x ad the choices fo y, the thee ae m choices fo (x, y). Fo Seveal: If thee ae i choices fo x i, i 1,,, the thee ae choices fo (x 1,, x ). 1 2 Example. Thee ae choices fo (wife, sack, cat). Example. Thee ae licese plates. Of these , 576, , 232, 000 have distict lettes ad digits (o epetitio). Pemutatios A pemutatio of legth is a list (x 1,, x ) with distict compoets (o epetitio); that is, x i x j whe i j. Examples. (1, 2, 3) is a pemutatio of thee elemets; (1, 2, 1) is a list, but ot a pemutatio Coutig Fomulas. Fom objects, lists of legth ad ( factos) () : ( 1) ( + 1) pemutatios of legth may be fomed. Examples: Thee ae thee digit umbes of which (10) list distict digits. Some Notatio. Recall () ( 1) ( + 1) positive iteges ad. Factoials: Whe, wite! () ( 1) 2 1. Covetios: () 0 1 ad 0! 1. Notes a). The book oly cosides. b). () 0 if >. c). If <, the! () ( )! Examples Example. A goup of 9 people may choose offices (P,VP,S,T) i (9) Example. 7 books may be aaged i 7! 5040 If thee ae 4 math books ad 3 sciece books, the thee ae 2 4! 3! 288 aagemets i which the math books ae togethe ad the sciece books ae togethe.

3 Combiatios A combiatio of size is a set {x 1,, x } of distict elemets. Two combiatios ae equal if they have the same elemets, possibly witte i diffeet odes. Example. {1, 2, 3} {3, 2, 1}, but (1, 2, 3) (3, 2, 1). Example. How may committees of size 4 may be chose fom 9 people? Choose offices i two steps: Choose a committee i?? Choose offices fom the committee i 4! Fom the Basic Piciple So, (9) 4 4!??.?? (9) 4 4! 126. Combiatios Fomula Fom 1 objects, 1! () combiatios of size may be fomed. Example. ( ) ! (9) Poof: Replace 9 ad 4 by ad i the example. Example: Bidge. A bidge had is a combiatio of 13 cads daw fom a stadad deck of N 52. Thee ae ( ) , 013, 559, such hads. Alteatively: Biomial Coefficiets!!( )!. The Biomial Theoem: Fo all < x, y <, (x + y) x y. Example. Whe 3, 0 (x + y) 3 x 3 + 3x 2 y + 3xy 2 + y 3. Poof. If (x + y) (x + y) (x + y) is expaded, the x y will appea as ofte as x ca be chose fom of the factos; i.e., i Recall: Biomial Idetities (x + y) 0 x y. Examples -a). Settig x y 1, 2. 0 b). Lettig x 1 ad y 1, ( 1) 0 fo 1. 0

4 Patitios AKA Divisios A Example Q: How may distict aagemets ca be fomed fom the lettes MISSISSIPPI? A: Thee ae 11 lettes which may be aaged i 11! 39, 916, 800 ( ) ways, but this leads to double coutig. If the 4 S s ae pemuted, the othig is chaged. Similaly, fo the 4 I s ad 2 P s. So, (*) the each cofiguatio of lettes times ad the aswe is 4! 4! 2! 1, ! 34, ! 4! 2! Patitios Defitios Let Z be a set with elemets. If 2 is a itege,the a odeed patitio of Z ito subsets is a list (Z 1,, Z ) whee Z 1,, Z ae mutually exclusive subsets of Z whose uio is Z; that is, ad Let Z i Z j if i j Z 1 Z Z. i #Z i, the umbe of elemets i Z i. The ad 1,, The Patitios Fomula Example. I the MISSISSIPPI Example, 11 positios, Z {1, 2,, 11} wee patitioed ito fou goups of sizes 1 4 I s 2 1 M s 3 2 P s 4 4 S s Example. I a bidge game, a deck of 52 cads is patitioed ito fou hads of size 13 each, oe fo each of South, West, Noth, ad East. Let,, ad 1,, be iteges fo which, 1, 1,, 0, If Z is a set of elemets, the thee ae ( )! : 1,, 1!! ways to patitio Z ito subsets (Z 1,, Z ) fo which #Z i i fo i 1,,. Example. ( ) 11 34, , 1, 2, 4 Def. Called multiomial coefficiets

5 The Numbe of Solutios If ad ae positive iteges, how may itege solutios to the equatios ae thee? 1,, Fist Wam Up Example. How may aagemets fom a A s ad b B s fo example, ABAAB)? Thee ae a + b a + b a b such, sice a aagemet is detemied by the a places occupied by A. The Numbe of Solutios Cotiued Secod Wam Up Example. Suppose 8 ad 4. Repeset solutios by o ad + by. Fo example, meas ooo oo ooo 1 3, 2 2, 3 0, 4 3. Note: Oly 1 3 s ae eeded. Thee ae as may solutios as thee ae ways to aage o ad. By the last example, thee ae solutios. A Geeal Fomula If ad ae positive iteges, the thee ae itege solutios to 1,, If, the thee ae 1 1 solutios with fo i 1,,. i 1 Combiatoics Summay Lists, pemuatatios, ad combiatios. The Basic Piciple Coutig Fomulas Lists Pemuatios () Combiatios ( ) Patitios ( 1,, ) Solutios ( ) + 1 1