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Permutations andPermutations andCombinationsCombinations



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PermutationPermutation

A A permutationpermutation is an arrangement of theis an arrangement of theobjects in a definite order.objects in a definite order.

For example, 3 blocks can be ordered 6 waysFor example, 3 blocks can be ordered 6 ways

There are n! permutations of n elementsThere are n! permutations of n elements

Easily proved using the Product ruleEasily proved using the Product rule



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rr--PermutationPermutation

rr--permutationspermutations are used to order subsets of are used to order subsets of distinct elementsdistinct elements

P(n,r) represents the rP(n,r) represents the r--permutations of a set of n distinctpermutations of a set of n distinctelementselements

P(n,r) = n(nP(n,r) = n(n--1)(n1)(n--2) « (n2) « (n--r+1) for rr+1) for r ee nn P(n,r) = n! / (nP(n,r) = n! / (n--r)!r)!

ExampleExample

There are 8 different candy to choose from. How manyThere are 8 different candy to choose from. How manydifferent ´mealsµ can you make by choosing 3 of them?different ´mealsµ can you make by choosing 3 of them?

P(n,r) = P(8,3) = 8*7*6P(n,r) = P(8,3) = 8*7*6



Permutations mainly occur where aPermutations mainly occur where arearrangement within a particular grouprearrangement within a particular groupwould change its nature into a differentwould change its nature into a different

selection.selection. Example: A group of people chosen to beExample: A group of people chosen to be

president , Vicepresident , Vice--President, Secretary, andPresident, Secretary, andTreasurer constitutes a different leadershipTreasurer constitutes a different leadership

group with each reassignment of titles withingroup with each reassignment of titles withinthe particular subgroup.the particular subgroup.

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rr--CombinationCombination

What if it does not matter which order you eatWhat if it does not matter which order you eatthe candy?...the candy?...

A A combinationcombination is anis an unorderedunordered arrangementarrangementof elements in a setof elements in a set

Example:Example: a 3a 3--combination from a set of 12combination from a set of 12

colored blockscolored blocks



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rr--CombinationCombination

AnAn rr--combinationcombination is a selection of ris a selection of runordered elements from a set (or simply aunordered elements from a set (or simply a

subset with r elements)subset with r elements) C(n,r) denotes the number of rC(n,r) denotes the number of r--combinations of acombinations of a

set of n elementsset of n elements

C(n,r)= P(n,r)/r! = n!/[(nC(n,r)= P(n,r)/r! = n!/[(n--r)!r!] = C(n,nr)!r!] = C(n,n--r)r)



Combinations occur in Problems regarding Combinations occur in Problems regarding card hands, committees, and othercard hands, committees, and othercircumstances where a rearrangement of acircumstances where a rearrangement of aparticular group is still considered the sameparticular group is still considered the samegroup.group.

Example: Four people sitting on a committeeExample: Four people sitting on a committeecan play musical chairs all day long, but it iscan play musical chairs all day long, but it is

still the same committee of four.still the same committee of four.

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Combinations with RepetitionCombinations with Repetition

Theorem:Theorem:

There are C( n + rThere are C( n + r -- 1, r) r1, r) r--combinations from acombinations from aset with n unique elements when repetition of set with n unique elements when repetition of

elements is allowed.elements is allowed. Previous Example:Previous Example:

n = 3 different fruitsn = 3 different fruits

r = 4 items to selectr = 4 items to select C( 3+4C( 3+4--1, 4) = C(6,4) = 6!/[(61, 4) = C(6,4) = 6!/[(6--4)!4!] = 154)!4!] = 15



Combinations with RepetitionCombinations with Repetition

ExampleExample

How many ways are there to select 3 bills from aHow many ways are there to select 3 bills from acash box containing $1, $5, $10, $20 and $50cash box containing $1, $5, $10, $20 and $50

bills?bills? SolutionSolution

$1$1 $5$5 $10$10 $20$20 $50$50

* * ** * *

** ****

1 2 3 4

n = 5

r = 3

C( 3+5-1, 3)

C( 7, 3)



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Combination with RepetitionCombination with Repetition

ExampleExample

How many ways can 6 balls be distributedHow many ways can 6 balls be distributedinto 9 different bins?into 9 different bins?

SolutionSolution

n = 9 unique binsn = 9 unique bins

r = 6 ballsr = 6 balls

C( 9+6C( 9+6--1, r ) = C(14, 6)1, r ) = C(14, 6)



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Counting permutations with repetition reliesCounting permutations with repetition relieson the multiplication ruleon the multiplication rule

ExampleExample::

How many strings of lengthHow many strings of length r r can be formedcan be formedfrom the English alphabet?from the English alphabet?

TheoremTheorem::

The number of rThe number of r--permutations of a set of npermutations of a set of nobjects with repetition is nobjects with repetition is nrr

Permutations with RepetitionPermutations with Repetition

r

abc abc abc abc « = 26*26*26 « 26

r

= 26r



There are 4 candidates for the post of aThere are 4 candidates for the post of alecturer in Mathematics and one is to belecturer in Mathematics and one is to beselected by votes of 5 men. The number of selected by votes of 5 men. The number of ways in which the votes can be given isways in which the votes can be given is

Sol: Each man can vote for any one of the 4Sol: Each man can vote for any one of the 4candidates and this can be done in 4 ways.candidates and this can be done in 4 ways.

5 men can vote in 45 men can vote in 455 = 1024 ways.= 1024 ways.

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rr--Permutations with RepetitionPermutations with Repetition

TheoremTheorem

The number of different permutations of nThe number of different permutations of n

objects, where there are nobjects, where there are n11

identical objects of identical objects of

type 1, ntype 1, n22 identical objects of type 2 « and nidentical objects of type 2 « and ntt

identical objects of type t, isidentical objects of type t, is

n!n!

nn11! n! n22! «n! «ntt!!



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Example: Which formula to use???Example: Which formula to use???

How many different strings can be made by How many different strings can be made by reordering the letters of the string SUCCESS?reordering the letters of the string SUCCESS?

SolutionSolution: 7!: 7!

3! X 2!X1!X1!3! X 2!X1!X1!

= 420= 420



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Permutations & CombinationsPermutations & Combinations

r-permutations

P(n,r)

r-combinations

C(n,r)

r-permutations

with repetition

combinations

with repetition

n!

(n-r)!

nr

C(n+r-1,r)

n!

(n-r)!r!

permutations

with repetition of

identical objects

n!

n1!n2! «nt!

For r elements selected from n distinct elements:

For n elements with n1, n2 « ntidentical objects For n unique elements in X withr selections from X

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