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The concept of a combination did not consider the order of the elements of the subset to be important. A permutation is a combination with the order of a selection from a group being important. For example, for the set { 1 , 2 , 3 , 4 , 5 , 6 } , the combination { 1 , 2 , 3 } would be identical to the combination { 3 , 2 , 1 } , but these two combinations are different permutations, because the elements in the set are ordered differently.

More formally, a permutation is an ordered list without repetitions, perhaps missing some elements.

This means that { 1 , 2 , 2 , 3 , 4 , 5 , 6 } and { 1 , 2 , 4 , 5 , 5 , 6 } are not permutations of the set { 1 , 2 , 3 , 4 , 5 , 6 } .

Now suppose you have these objects:

1, 2, 3

Here is a list of all permutations of all three objects:

1 2 3; 1 3 2; 2 1 3; 2 3 1; 3 1 2; 3 2 1.

Counting permutations

Let S be a set with n objects. Permutations of r objects from this set S refer to sequences of r different elements of S (where two sequences are considered different if they contain the same elements but in a different order). Formulas for the number of permutations and combinations are readily available and important throughout combinatorics.

It is easy to count the number of permutations of size r when chosen from a set of size n (with r n ).

  1. Select the first member of the permutation out of n choices, because there are n distinct elements in the set.
  2. Next, since one of the n elements has already been used, the second member of the permutation has ( n - 1 ) elements to choose from the remaining set.
  3. The third member of the permutation can be filled in ( n - 2 ) ways since 2 have been used already.
  4. This pattern continues until there are r members on the permutation. This means that the last member can be filled in ( n - ( r - 1 ) ) = ( n - r + 1 ) ways.
  5. Summarizing, we find that there is a total of
    n ( n - 1 ) ( n - 2 ) . . . ( n - r + 1 )
    different permutations of r objects, taken from a pool of n objects. This number is denoted by P ( n , r ) and can be written in factorial notation as:
    P ( n , r ) = n ! ( n - r ) ! .

For example, if we have a total of 5 elements, the integers { 1 , 2 , 3 , 4 , 5 } , how many ways are there for a permutation of three elements to be selected from this set? In this case, n = 5 and r = 3 . Then, P ( 5 , 3 ) = 5 ! / 7 ! = 60 ! .

Khan academy video on probability - 2

Show that a collection of n objects has n ! permutations.

  1. Proof: Constructing an ordered sequence of n objects is equivalent to choosing the position occupied by the first object, then choosing the position of the second object, and so on, until we have chosen the position of each of our n objects.

  2. There are n ways to choose a position for the first object. Once its position is fixed, we can choose from ( n - 1 ) possible positions for the second object. With the first two placed, there are ( n - 2 ) remaining possible positions for the third object; and so on. There are only two positions to choose from for the penultimate object, and the n t h object will occupy the last remaining position.

  3. Therefore, according to the fundamental counting principle, there are

    n ( n - 1 ) ( n - 2 ) . . . 2 × 1 = n !

    ways of constructing an ordered sequence of n objects.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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