Counting Rules
Rules of Filling
the Place
Let us pay attention to the following
introduction of the rules of filling the place.
Example 1
In a-one hundred
meters run champion,four participants had passed to thr final round,there are A
(Adi), B (Banu), C (Candra),and D (Doni).Inthe last round,two prizes for he two
winners will be presented.How many arrange of winner will be appeared at the
end of the race ?
Answer :
The first nd
second winner that probably appeared at the end of the race can be arranged as
follow :AB,AC,AD,BA,BC,BD,CA,CB,CD,DA,DB,and
DC.The process is determining the
number of winner arrangement the rules of follows :
First : Every
participant has an opportunity to be the first winner
Second: As one
participant already go throught the finish line,there are still there other
participants who has opportunity to be a second winner.
Therefore,there
are 
ways to arrange two ossible winners out of
four participant.
ways to arrange two ossible winners out of
four participant.
From the
explanation,the following conclusion can be obtained.
If there is k places
provided to be filled in,where :
= number of ways to fill the first place
= number of ways to fill the first place
= number of ways to fill the second place,ater
the first place had already filled in
= number of ways to fill the third place,after
the first and second place had already filled in
= number of ways to fill the k-order
place,after the previous places had already filled in
Then,the number of ways
in arranging k-places to be filled in is
The above is called
rules of filling places or multiplication rule.
Examples of Question
1. Amaliah
has 4 blazers,2 trousers,and 3 shoes.How many ways for her to completely
dressed up ?
Answer :
Answer :
Amaliah has 4 option to
wear blazer,2 optio to wear trousers,and 3 option for shoes.
Blazers
|
Trousers
|
Shoes
|
4
|
3
|
2
|
So,there are 4 x
3 x 2 =24 ways for Amaliah to completely dressed up.
2. A police number plates starting
with the letter H semarang followed by four
digits with the first number
should not be terminated nol.Dan hurf last
two letters with
the letter A.
After the car license plate number of the model should be modified?
After the car license plate number of the model should be modified?
Answer
:
License
Plate
H
|
X
|
X
|
X
|
X
|
X
|
A
|
H
|
Be filled the numbers 1-9
|
Be filled the numbers 0-9
|
Be filled the numbers 0-9
|
Be filled the numbers 0-9
|
be filled
letters AZ
|
only be
filled by the letter A
|
1
|
9
|
10
|
10
|
10
|
26
|
1
|
Suppose a license plate consists of 7 boxes, then:
The first letter in the box pertma can be printed in a way (because it can only be filled by the letter H)
The first digit in the second box can be printed in 9 ways (because the first box should not be number 0)
The second number in the third box can be printed in 10 ways (as it can be filled the numbers 0-9)
The third digit in the third box can be printed in 10 ways (as it can be filled in the numbers 0-9)
The fourth digit in the third box can be printed in 10 ways (as it can be filled in the numbers 0-9)
The second letter in the sixth box can be printed in 26 point (since it can be filled letters AZ)
The third in the seventh letter of the brain can be printed in one way (because it can only be filled by the letter A)
So
many different license
plates that can be printed is 1
x 9 x 10 x 10 x 10 x 26 x 1 = 234 000. Because
every single license
plate for only one car number plate then the
model must be changed after the car to
234 000.
3. How many
papers are to be
provided, if every paper written on a 3
digit number formed from the five the
numbers 1,3,5,7,9, if:
a) Repetition is not allowed
b) Repetition is allowed
a) Repetition is not allowed
b) Repetition is allowed
Answer
:
Suppose
there are three boxes to present any numbers.
First box
|
Second box
|
Third box
|
5
|
4
|
3
|
a)
The first box can be filled in 5 ways, because repetition is not allowed then the second box and three boxes of each can be filled with 4 and 3 ways.
So many of his numbers can be formed there are 5 x 4 x 3 = 60 numbers. So many of his papers are to be provided there are 60 pieces.
The first box can be filled in 5 ways, because repetition is not allowed then the second box and three boxes of each can be filled with 4 and 3 ways.
So many of his numbers can be formed there are 5 x 4 x 3 = 60 numbers. So many of his papers are to be provided there are 60 pieces.
First box
|
Second box
|
Third box
|
5
|
5
|
5
|
b) Because
repetition is allowed the box first, second and
third can be filled in 5 ways,
so the number of numbers that form there are 5 x
5 x 5 = 125 numbers.
So many of his papers are to be provided there is 125 sheets.
So many of his papers are to be provided there is 125 sheets.
c) Of the five points 0, 3, 4, 5, 7 will be formed of a number consisting
of 4 digits.
How many numbers can
be formed if
the which:
a) the numbers may be repeated
b) the numbers should not be repeated
a) the numbers may be repeated
b) the numbers should not be repeated
solution:
a) The first number Thousands to choose from as four possibilities, namely 3, 4, 5 or 7. Number 0 may not be the first number of Because It would cause the number formed only consist of 3 digits. Because the numbers may be repeated Hundreds, and TENS units of each of the selected individual can be possible..
a) The first number Thousands to choose from as four possibilities, namely 3, 4, 5 or 7. Number 0 may not be the first number of Because It would cause the number formed only consist of 3 digits. Because the numbers may be repeated Hundreds, and TENS units of each of the selected individual can be possible..
thousands
|
hundreds
|
tens
|
unit
|
4
|
5
|
5
|
5
|
3,4,5,7
|
0,3,4,5,7
|
0,3,4,5,7
|
0,3,4,5,7
|
The
number of number 5 is formed there are 4 x 5 x 5 x 5 = 500 numbers
b)
The first number
Thousands to choose
from as four
ways. Because it
should not be
repeated while a
figure been used
in the first number
is the number of Airways to choose the second
number was only 4
ways. (Suppose the
first number 3 then the option is selected in the second number is 0,
4, 5atau 7.)
The number of
options on three
numbers, there are
three Airways and
there are many
options on the number to four two ways.
3. The amount of numbers can be formed there are 4 x 4 x 3 x 2 = 96 numbers) The first number Thousands to choose from as four ways. Because it should not be repeated while a figure been used in the first number is the number of Airways to choose the second number was only 4 ways. (Suppose the first number 3 then the option is selected in the second number is 0, 4, 5atau 7.) The number of options on three numbers, there are three Airways and there are many options on the fourth two ways.
3. The amount of numbers can be formed there are 4 x 4 x 3 x 2 = 96 numbers) The first number Thousands to choose from as four ways. Because it should not be repeated while a figure been used in the first number is the number of Airways to choose the second number was only 4 ways. (Suppose the first number 3 then the option is selected in the second number is 0, 4, 5atau 7.) The number of options on three numbers, there are three Airways and there are many options on the fourth two ways.
thousands
|
hundreds
|
tens
|
unit
|
4
|
4
|
3
|
2
|
3,4,5,7
|
0,3,4,5,
|
0,3,4
|
0,3
|
The
amount of numbers can be formed there are 4 x 4 x 3 x 2 = 96 numbers
4. Denny
will form a 3
digit even number that the
numbers taken from the 2, 3, 4, 5, 6, 7, 8.
How many numbers can be formed if:
a) the numbers may be repeated
b) the numbers should not be repeated
How many numbers can be formed if:
a) the numbers may be repeated
b) the numbers should not be repeated
solution:
a) The first number Hundreds to choose from as seven possibilities. The second number can be selected also from as 7 possibility. Because the number is even then the number of units can be selected only four possibilities is 2, 4, 6 or 8.
a) The first number Hundreds to choose from as seven possibilities. The second number can be selected also from as 7 possibility. Because the number is even then the number of units can be selected only four possibilities is 2, 4, 6 or 8.
hundreds
|
tens
|
unit
|
7
|
7
|
4
|
The
which are formed many numbers
there are 7
x 7 x 4 = 196 numbers.
b)
A number is
said to be even
or odd enough
to see the
units. Because the numbers even then
the selection is
first performed on
the unit numbers.
Unit numbers can
be selected from the four possibilities,
namely 2, 4,
6 or 8. Figures 6 Dozens
to choose from
Hundreds of numbers
can be selected
from 5 possibilities.
hundreds
|
tens
|
unit
|
5
|
6
|
4
|
The
amount of numbers can be formed there are 5 x 6 x 4 = 120 numbers
Question
1. Of the
seven numbers 1,
3, 5, 6, 7, 8, 9, Furkan will form
a three-digit numbers and more dari600. How
many numbers can be formed if:
a) the numbers may be repeated
b) the numbers should not be repeated
a) the numbers may be repeated
b) the numbers should not be repeated
2. Hansen
has been tasked
to form a three-digit numbers less
than 500 the
numbers are 2, 3,
4, 5, 6, 7 or 9. How many numbers can
be formed if:
a) the numbers may be repeated
b) the numbers should not be repeated
a) the numbers may be repeated
b) the numbers should not be repeated
3. Of the seven numbers
1, 3, 4, 5, 6, 8, 9 will be formed of a number 3 and number
over 500. How
many even numbers can be formed That if the numbers may be repeated
4. With the numbers 2, 3, 4, 5, 6, and 7 are made of numbers consisting of
three different numbers,
how many even
numbers can be
made different?
Completion: Three points means the first
three boxes are
made,
namely: the Hundreds,
TENS, units
5. From the figures 2.3,
5, 6, 7, and 9 is created consisting
of three different numbers.
The number of small numbers made more than 400 number is ....
The number of small numbers made more than 400 number is ....
6. Provided
the numbers 3,
4, 5, 6, 7, and 8. How many
integers are to
be formed if the
number is composed
of four digit
numbers of funds.
each not contain the same number; b. eachnumber may contain the same number.
each not contain the same number; b. eachnumber may contain the same number.
7. Someone wants to
make license plates consisting of 4 digits,
available when the numbers 1, 2, 3, 4, 5 and the number plate there should not be
the same figure. How many license plates can be made?
available when the numbers 1, 2, 3, 4, 5 and the number plate there should not be
the same figure. How many license plates can be made?
8. Of the card number 4, 5,
6, 7, and 8
is made of numbers of three figures
different. Determine the number of numbers less
a. of 500 b. of 600
different. Determine the number of numbers less
a. of 500 b. of 600
9. Of the
6 students will take
a picture image. If
the decision of each
images consist of two people, what is the number of ways making
mungklin images that happen?
images consist of two people, what is the number of ways making
mungklin images that happen?
10.
What
is the number of numbers
that consists of 3
digits can be arranged
of the numbers 1, 3, 5 and 7 if the numbers:
a. must not appear again
b. must not appear again
of the numbers 1, 3, 5 and 7 if the numbers:
a. must not appear again
b. must not appear again
Answer
Key
1. solution:
A three digit number is said to more than 600 if the digits are Hundreds of at least 6.
a) The first number Hundreds to choose from as four possibilities, namely 6, 7, 8 or 9. The figure also both be selected from seven possibilities. Unit numbers can also be selected from 7 possibilities.
A three digit number is said to more than 600 if the digits are Hundreds of at least 6.
a) The first number Hundreds to choose from as four possibilities, namely 6, 7, 8 or 9. The figure also both be selected from seven possibilities. Unit numbers can also be selected from 7 possibilities.
hundreds
|
tens
|
unit
|
4
|
7
|
7
|
That
form many numbers
there are 4
x 7 x 7 = 196 numbers
b)
The first number
Hundreds to choose
from as four
possibilities, namely 6, 7, 8 or 9. Puluhand apat selected number
of 6 means
the unit can
be selected while
the rate of
5 possible.
hundreds
|
tens
|
unit
|
4
|
6
|
5
|
The
amount of numbers can be formed there are 4 x 6 x 5 = 120 numbers
2. Solution:
A 3 digit number said to be less than 500 if the Hundreds digit is less than 5.
a) The first number Hundreds to choose from as three possibilities, namely 2, 3 or 4. The second number can be selected of 7 possible. Unit numbers can also be selected from 7 possibilities.
That form many numbers there are 3 x 7 x 7 = 147 numbers.
A 3 digit number said to be less than 500 if the Hundreds digit is less than 5.
a) The first number Hundreds to choose from as three possibilities, namely 2, 3 or 4. The second number can be selected of 7 possible. Unit numbers can also be selected from 7 possibilities.
That form many numbers there are 3 x 7 x 7 = 147 numbers.
hundreds
|
tens
|
unit
|
3
|
7
|
7
|
b) The first number Hundreds to choose from as three possibilities, namely 2, 3 or 4. Puluhand apat selected number of 6 means the unit can be selected while the rate of 5 possible.
The amount of numbers can be formed there are 3 x 6 x 5 = 90 numbers
hundreds
|
tens
|
unit
|
3
|
6
|
5
|
3. solution:
The first number Hundreds to choose from as four possibilities, namely 5, 6, 8 or 9. Both numbers can be selected from seven possibilities. Figures 3 units can be selected from any number formed possibilities.
Many there are 4 x 7 x 3 = 84 numbers.
The first number Hundreds to choose from as four possibilities, namely 5, 6, 8 or 9. Both numbers can be selected from seven possibilities. Figures 3 units can be selected from any number formed possibilities.
Many there are 4 x 7 x 3 = 84 numbers.
hundreds
|
tens
|
unit
|
4
|
7
|
3
|
4. Completion: Three points means
the first three boxes
are made,
namely: the Hundreds,
TENS, units
hundreds
|
tens
|
unit
|
4
|
5
|
3
|
So, many even numbers are made up of three numbers is (4 x 5 x 3) = 60 numbers
5. Answer:
Hundreds: The place is filled by Hundreds 2dan can only figure 3 for a small number that must be formed from 400 to
n (1) = 2
Dozens: The place can only be filled by Dozens of 5 digits (selectable) since the place has hundreds of numbers can use to fill That
n (2) = 5
Unit: Outdoor units can only be filled by a 4 digit (selection) Because the numbers have been used for Decades to fill the place.
n (3) = 4
Using the multiplication rule, the number of numbers consisting of three numbers is smaller than 400 .....
n (1) x n (2) x n (3) = 2 x 5 x 4 = 40
Hundreds: The place is filled by Hundreds 2dan can only figure 3 for a small number that must be formed from 400 to
n (1) = 2
Dozens: The place can only be filled by Dozens of 5 digits (selectable) since the place has hundreds of numbers can use to fill That
n (2) = 5
Unit: Outdoor units can only be filled by a 4 digit (selection) Because the numbers have been used for Decades to fill the place.
n (3) = 4
Using the multiplication rule, the number of numbers consisting of three numbers is smaller than 400 .....
n (1) x n (2) x n (3) = 2 x 5 x 4 = 40
6. solution:
a) Because each number can not take the same figure, the first digit (in Thousands) can be selected with 6 way, that the figures 3, 4, 5, 6, 7, and 8.
The second digit (the Hundreds) can be selected with the 5 way for a number telahdipilih as the first digit. Then, the third digit (for TENS) can be selected with 4 way and the fourth digit (the units) in three ways. This case can be described by the Following Table
a) Because each number can not take the same figure, the first digit (in Thousands) can be selected with 6 way, that the figures 3, 4, 5, 6, 7, and 8.
The second digit (the Hundreds) can be selected with the 5 way for a number telahdipilih as the first digit. Then, the third digit (for TENS) can be selected with 4 way and the fourth digit (the units) in three ways. This case can be described by the Following Table
thousands
|
hundreds
|
tens
|
unit
|
6
|
5
|
4
|
3
|
So, a lot of numbers consisting of 4
different numbers That can be formed from the numbers 3, 4, 5, 6, 7, and 8
there are 6 × 5 × 4 × 3 = 360 number
b) Since each number should contain the same number of each there are six way to occupy the first place, second, third, and fourth.
Thus, the number of numbers can be formed That there are 6 × 6 × 6 × 6 = 1296 numbers.
thousands
|
hundreds
|
tens
|
unit
|
6
|
6
|
6
|
6
|
7. completion:
To answer these questions, we use the "fill in the blank"
as shown in the following chart.
To answer these questions, we use the "fill in the blank"
as shown in the following chart.
a
|
b
|
c
|
d
|
Created
4 pieces of empty boxes is a box (a), (b),(c) and (d) because the
number of vehicles that consists of
4 numbers
a
|
b
|
c
|
d
|
5
|
Box
(a) may be filled
the numbers 1, 2, 3, 4, or 5.So there are 5 ways
a
|
b
|
c
|
d
|
5
|
4
|
Box
(b) can only be filled in figures 5-1 = 4 ways as a way already loaded
in box (a).
a
|
b
|
c
|
d
|
5
|
4
|
3
|
Box
(c) can only be filled in figures 5-2 = 3 ways because the two
ways already loaded
in box (a)
and (b).
a
|
b
|
c
|
d
|
5
|
4
|
3
|
2
|
Box
(d) can only be filled in figures 5-3 = 2 ways as 3 ways
already loaded in
box (a), (b),
and
(c).
(c).
So,
the police can make
the license plates of vehicles
as much as 5 x 4
x 3 x 2 = 120
license plates.
license plates.
8. solution:
a) Because the numbers are less than 500 then the
number of hundreds only
filled by one point, the number 4. One of the possible arrangements can be seen
in the figure below.
Tens digit and the unit can be charged by the numbers 5, 6, 7, and 8. This means you
must choose two numbers of 4 digits, ie:
filled by one point, the number 4. One of the possible arrangements can be seen
in the figure below.
Tens digit and the unit can be charged by the numbers 5, 6, 7, and 8. This means you
must choose two numbers of 4 digits, ie:
4P2 = 
So, there are 12 ways to make numbers less than 500
So, there are 12 ways to make numbers less than 500
b) Because
the numbers were less
than 600 then the number is only filled hundreds
by two numbers, the numbers 4 and 5.
4
numbers
of tens and units
can be filled by the
numbers 5, 6, 7, and 8 (select
2 from
4 elements).
5
number
of tens and units
can be filled by the
numbers 4, 6, 7, and 8 (select
2 from
4 elements).
2 x 4P2 = 2 x
So, there are 24 ways to make numbers less than 600.
by two numbers, the numbers 4 and 5.
4
numbers
of tens and units
can be filled by the
numbers 5, 6, 7, and 8 (select
2 from4 elements).
5
number
of tens and units
can be filled by the
numbers 4, 6, 7, and 8 (select
2 from4 elements).
2 x 4P2 = 2 x
So, there are 24 ways to make numbers less than 600.
9.
Answer:
As an illustration made 2 points for 2 people taken aka
picture, which is where I and II below.
As an illustration made 2 points for 2 people taken aka
picture, which is where I and II below.
The place I: can be occupied by a student of 6 students, so
there are 6 ways
I II III
II live in: it can be occupied by students from the five students, for which one
students are already in place I, so there are 5 ways.
So many ways there are photo shooting: 6 x 5 = 30 ways
I II III
II live in: it can be occupied by students from the five students, for which one
students are already in place I, so there are 5 ways.
So many ways there are photo shooting: 6 x 5 = 30 ways
I
|
II
|
6
|
5
|
10.
Answer:
as an overview is provided of three places for the following three points :
as an overview is provided of three places for the following three points :
a) If those numbers may appear repetitive. Place
I, II, and III
numbers can be occupied respectively by 4 ways, so
any number of numbers = 4 x 4 x 4 = 64 ways
numbers can be occupied respectively by 4 ways, so
any number of numbers = 4 x 4 x 4 = 64 ways
I
|
II
|
III
|
4
|
4
|
4
|
b) If those numbers should not appear again.
Place I, II and III
numbers can be assigned consecutive 4-way, 3 way, and 2
ways. So there are numbers banaknya 4 x 3 x 2 = 24 ways
numbers can be assigned consecutive 4-way, 3 way, and 2
ways. So there are numbers banaknya 4 x 3 x 2 = 24 ways
I
|
II
|
III
|
4
|
3
|
2
|
Foreword
This is a mathematical module handle as well as a source of teaching and learning process students IKIP PGRI Semarang.
Assessment of each subject matter based on the sequence of learning indicators,so that it can be followed by all students from one material to another material. In addition,This module is compiled with simpler language in order to better easily be studied by students without the guidance of a maximum of Master Teachers. The author is fully aware that this module is still far from perfect. Therefore, criticism and suggestions relevant to improvement.
This module is the author expected. The author would also like to thank all those who have helped in the preparation of this module.
This module may be useful and can provide added value tointernal and external educational progress in IKIP PGRI Semarang.
Semarang, July 14, 2012
compiler
This is a mathematical module handle as well as a source of teaching and learning process students IKIP PGRI Semarang.
Assessment of each subject matter based on the sequence of learning indicators,so that it can be followed by all students from one material to another material. In addition,This module is compiled with simpler language in order to better easily be studied by students without the guidance of a maximum of Master Teachers. The author is fully aware that this module is still far from perfect. Therefore, criticism and suggestions relevant to improvement.
This module is the author expected. The author would also like to thank all those who have helped in the preparation of this module.
This module may be useful and can provide added value tointernal and external educational progress in IKIP PGRI Semarang.
Semarang, July 14, 2012
compiler
Bibliography
Dra.Arief
Agoestanto, M.Si. 2010 . Pengantar
Probabilitas (Universitas Negeri Semarang).Semarang.
Anggota
Ikapi. 2009. Mathematics For Senior High
School Year XI.Jakarta:Yudhistira
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