Permutations and Combinations ppt

  • View
    223

  • Download
    0

Embed Size (px)

Text of Permutations and Combinations ppt

  • 7/24/2019 Permutations and Combinations ppt

    1/26

    11

    Permutations andPermutations and

    CombinationsCombinationsCS/APMA 202CS/APMA 202

    Rosen section 4.3Rosen section 4.3Aaron BloomfieldAaron Bloomfield

  • 7/24/2019 Permutations and Combinations ppt

    2/26

    2

    Permutations vs. CombinationsPermutations vs. Combinations

    Both are a!s to count the "ossibilitiesBoth are a!s to count the "ossibilities

    #he difference beteen them is hether order#he difference beteen them is hether ordermatters or notmatters or not

    Consider a "o$er hand%Consider a "o$er hand% AA&' ()' *+' ,0-' -&' ()' *+' ,0-' -

    s that the same hand as%s that the same hand as% -' ,0-' *+' ()'-' ,0-' *+' ()'AA&&

    oes the order the cards are handed outoes the order the cards are handed outmatter1matter1 f !es' then e are dealin ith "ermutationsf !es' then e are dealin ith "ermutations f no' then e are dealin ith combinationsf no' then e are dealin ith combinations

  • 7/24/2019 Permutations and Combinations ppt

    3/26

    3

    PermutationsPermutations

    A "ermutation is an ordered arranement of theA "ermutation is an ordered arranement of theelements of some setelements of some set SS etet SS 5a' b' c6 5a' b' c6

    c' b' a is a "ermutation of Sc' b' a is a "ermutation of S b' c' a is ab' c' a is a differentdifferent "ermutation of S"ermutation of S

    AnAn rr7"ermutation is an ordered arranement of7"ermutation is an ordered arranement of rrelements of the setelements of the set

    AA&' ()' *+' ,0-' - is a (7"ermutation of the set of&' ()' *+' ,0-' - is a (7"ermutation of the set ofcardscards

    #he notation for the number of#he notation for the number of rr7"ermutations%7"ermutations%PP88nn''rr99

    #he "o$er hand is one of P8(2'(9 "ermutations#he "o$er hand is one of P8(2'(9 "ermutations

  • 7/24/2019 Permutations and Combinations ppt

    4/26

    4

    PermutationsPermutations

    :umber of "o$er hands 8( cards9%:umber of "o$er hands 8( cards9% PP8(2'(9 (2;(,;(0;4

  • 7/24/2019 Permutations and Combinations ppt

    5/26

    (

    rr7"ermutations [email protected]"le7"ermutations [email protected]"le

    o man! a!s are there for ( "eo"le ino man! a!s are there for ( "eo"le in

    this class to ive "resentations1this class to ive "resentations1

    #here are 2* students in the class#here are 2* students in the class P82*'(9 2*;2?;2(;24;23

  • 7/24/2019 Permutations and Combinations ppt

    6/26

    ?

    Permutation formula "roofPermutation formula "roof

    #here are#here are nn a!s to choose the firsta!s to choose the first

    elementelement nn7, a!s to choose the second7, a!s to choose the second

    nn72 a!s to choose the third72 a!s to choose the third

    nn77rrD, a!s to choose theD, a!s to choose the rrththelementelement

    B! the "roduct rule' that ives us%B! the "roduct rule' that ives us%

    PP88nn''rr9 9 nn88nn7,987,98nn72987298nn77rrD,9D,9

  • 7/24/2019 Permutations and Combinations ppt

    7/26

    *

    Permutations vs.Permutations vs. rr7"ermutations7"ermutations

    rr7"ermutations% Choosin an ordered (7"ermutations% Choosin an ordered (

    card hand iscard hand is PP8(2'(98(2'(9 Ehen "eo"le sa! F"ermutationsG' the! almostEhen "eo"le sa! F"ermutationsG' the! almost

    ala!s meanala!s mean rr7"ermutations7"ermutations

    But the name can refer to bothBut the name can refer to both

    Permutations% Choosin an order for all (2Permutations% Choosin an order for all (2

    cards iscards is PP8(2'(29 (28(2'(29 (2 #hus'#hus' PP88nn''nn9 9 nn

  • 7/24/2019 Permutations and Combinations ppt

    8/26

    =

    Rosen' section 4.3' Huestion 3Rosen' section 4.3' Huestion 3

    o man! "ermutations of 5a' b' c' d' e' f' 6o man! "ermutations of 5a' b' c' d' e' f' 6end ith a1end ith a1 :ote that the set has * elements:ote that the set has * elements

    #he last character must be a#he last character must be a #he rest can be in an! order#he rest can be in an! order

    #hus' e ant a ?7"ermutation on the set 5b' c'#hus' e ant a ?7"ermutation on the set 5b' c'

    d' e' f' 6d' e' f' 6P8?'?9 ? *20P8?'?9 ? *20

    Eh! is it not P8*'?91Eh! is it not P8*'?91

  • 7/24/2019 Permutations and Combinations ppt

    9/26

Search related