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7/24/2019 Permutations and Combinations ppt

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11

Permutations andPermutations and

CombinationsCombinationsCS/APMA 202CS/APMA 202

Rosen section 4.3Rosen section 4.3Aaron BloomfieldAaron Bloomfield

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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

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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

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4

PermutationsPermutations

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

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(

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

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?

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

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*

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

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=

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

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