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Nguyễn Như Quỳnh
Giáp T.Thục Trinh
Nguyễn Vân Trang
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Permutation with repetition
Discrete mathematics
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LECTURERS : TRAN NGUYEN AN
Permutation with repetition
PERMUTATION WITH REPETITION
AND SOME RELATED LESSONS
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Discrete
mathematics
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Discrete mathematics
CONTENT 1:
PREPARING KNOWLEDGE
PERMUTATIONS
CONTENT 2:
PERMUTATION WITH REPETITION
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Mã Sinh Viên CONTENT
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Viên
EXERCISES
CONTENT 4:
SOME APPLICATIONS OF PERMUTATION
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PERMUTATION
PERMUTATION
PERMUTATION WITH REPETITION AND SOME RELATED LESSONS
PERMUTATION WITH REPETITION AND SOME RELATED LESSONS
Discrete mathematics
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CONTENT 1:
PREPARING KNOWLEDGE
PERMUTATIONS
CONTENT 2:
PERMUTATION WITH REPETITION
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:
Mã Sinh Viên CONTENT
Mã3 Sinh
Viên
EXERCISES
CONTENT 4:
SOME APPLICATIONS OF PERMUTATION
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PERMUTATION
PERMUTATION
PERMUTATION WITH REPETITION AND SOME RELATED LESSONS
PERMUTATION WITH REPETITION AND SOME RELATED LESSONS
Discrete mathematics
Permutation with repetition
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Content 1
Content 2
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DESCRIPTION
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Members of the group
Discrete mathematics
Permutation with repetition
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3. Corollary:
The number of bijections from a set of elements to a set of elements is .
Contents 3
Contents 4
I. Permutations:
II. Generalize Permutations:
1. Definition:
Definition: of a set of distinct objects is an ordered arrangment of these objects. An
A1.permutation
ordered arrangement of r elements (or an ordered r-arrangement ) of a set is called an rAn ordered r-arrangement with repetition of the elements of the set with n elements is
permutation. The number of r-permutations of a set with n elements is denoted by or . If ,
also
called by
an . r-permutations of a set with n elements when repetition is allowed. The
we
denoted
2. Theorem:
number of r-permutations of a set with n elements when repetition is allowed is
- If is a positive integer and r is an integer with , then there are
denoted by .
r-permutations of a set with n distinct elements.
2. Theorem:
- The number of permutations of n elements:
The number of r-permutations of a set of n objects with repetition allowed is
Contents 2
CONTENT 1:PREPARING KNOWLEDGE: PERMUTATIONS
I. Definition of permutation with repetition
Contents 1
CONTENT 2: PERMUTATION WITH REPETITION
For n elements, including n1 element x1, n2 elements x2, ..., nk element xk
The number of all repeating permutations of n elements above is:
Contents 4
permutation of the given element n.
Contents 3
(n1+n2+...+nk=n). Each way of arranging n that element into n position is called a repeating
Contents 1
Contents 2
Example 2:
3: Given
With the
{0;3;1;5;2;6;3;9}.4;From
5} how
manymany
numbers
of 8 have
digits7can
be so
set digits
A= { 1;
A, how
numbers
digits
Example
1: From
X={1;2;3;4;5;6;7;8}.
numbers
have
11exactly
digits
so
made,
which
the
number
is present
3 How
times,
eachnatural
other
digit
is present
that
theofnumber
1 set
appears
2 1times;
the number
6many
appears
twice;
other
numbers
appear
once
andonce
thatand
number
is divisible
by 5?
exactly
number
divisible
5. are present once?
that the
number
1this
is present
4istimes,
otherbydigits
Solution.
Solution.
Solution.
that the
8-digit number
. in which the number 1 is present exactly 3 times;
⇒ Suppose
The number
of numbers
with 7 is:
digits
the
number
with
digits satisfied
the
first
is:number 1 is present 4 times, the
Since
the
number
divisible
by
5, 11
a8 =
0 or
5.
Each
way
of 3;4
making
a7 number
has
digits
soarticle
that the
theCall
digits
0;
2;
areispresented
exactly
once:
Because
this
number
is divided by 5, a7 = 5.
+
Case
1.
If
a
=
5.
840
–
120
=
720
numbers
8
other digits present 1 are a repeating permutation of 11 elements.
The2.problem
becomes
the becomes
calculation
of thea number
6-digit
numbers
created
from 1
++
Case
If a8on,
= 0the
. problem
From
now
finding
number of
with
7 digits
in which
the number
settime;
{ 1;3to
3;
6;problem
9}
so that
the0;number
1 appears
2 times;
number
6 appears
2 times 1 is
According
the
rules
of
repeating
there
are7the
isthat
present
times;
the
digits
2; finding
3;permutation,
4 are
exactly
once.
Atthe
the
becomes
apresented
number
with
digits
in which
the number
and
9 3only
appeared
once.
We 3;
have:
presented
times;
digits 2;
3; 4:5 is presented exactly once.
According
to the rules of repeat permutation
there are:
numbers
We
have:
If the first digit is 0; we’ll find the number of numbers with 6 digits in which the number 1
There
arewe
180can
the2;problem.
is presented
3 numbers
times
andthat
thesatisfy
number
3; 4 appear once.
Therefore,
make:
We840
have
: numbers in total.
720+
= 1560
numbers
Contents 4
CONTENT 3 : EXERCISES
Contents 1
Contents 2
3.
number
ofaavailable
2.
Working
outthe
a way
to human
win
lottery.
1.Finding
Protein out
formation
in
body. phone numbers.
No
two play
phonea numbers
are supposed
be alike.
If you
lottery game,
you maytowant
to know what your odds are for winning the
Proteins
are formed afrom
that the
contain
an of
arrangement
of amino
Using
phonestring-like
company structures
can determine
number
unique telephone
prize. permutations,
acids.
numbers
it canifissue
based on
the say
number
use.
For example,
the lottery
rules
you format
can winit ifwants
you to
pick
four digits that match (e.g.,
The
amino
is
important
because
this
will0determine
whether
If
thesequence
phone
should
all consist
of 10
digits
and
can
use
toby9, using
an example
of the
one
1111,
9999, numbers
or of
5555),
youacids
can
work
out
your
odds
for
winning
a permutation
protein
will function
not.
such
number
could be:or5653278065
calculation.
Defective
or
missing
proteins
cause
in ways
people
suchslot
as or
sickle
We
canslot
work
out
the
total
numbers
that
canserious
beinavailable
in this
way:(since
each
digit0cell
Each
(digit
position)
can can
be occupied
10diseases
different
we
have
to 9
anemia.
position
digits). can be occupied by any of the 10 digits (0 to 9).
For
example,
insulin
is atotal
protein
found
humans.4-digit
Its role
is to control
of =
Since
repetition
is allowed,
thisnumbers
works
out
So, we
can calculate
the
ofinto:
possible
numbers
as 10 xthe
10amount
x 10 x 10
sugar
the numbers.
body so that it is neither
high nor too low.
10x10x10x10x10x10x10x10x10x10
=10¹⁰too
numbers.
10,000around
different
Insulin
is made
of 51amino
acids
arranged
in athe
specific
sequence
orworld
permutation.
These
a lotwe
ofupcount
numbers
easily
outnumber
population
of the
(5 or 6
From are
there,
thethat
total
of winning
numbers
(0000,
1111,2222,
3333……9999)
Any
that deviates from this normal sequence makes this protein
billion
people).
whichrearrangement
are 10.
dysfunctional
and
causes
diseases
suchalso
as diabetes.
However,
because
the phone
include
for area
codes,
So, there are
10
numbers
thatnumbers
can win
and
the totaldigits
numbers
you can
draw are 10.000.
The
body has
ahave
mechanism
to ensure
that the
thisright
sequence
is followed
and =0.001.
the correct
they
generate
fewer
numbers
than
this.
Therefore,
you
a probability
of picking
sequence
of 10/10,000
is formed.
protein
In percentage
terms, you have a 0.1% (0.001 x 100) chance of winning.
Contents 3
CONTENT 4: SOME APPLICATIONS OF PERMUTATION
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