# 10.2 Permutations and applications

 Page 1 / 2

## Permutations

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 $\left\{1,2,3,4,5,6\right\}$ , the combination $\left\{1,2,3\right\}$ would be identical to the combination $\left\{3,2,1\right\}$ , 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 $\left\{1,2,2,3,4,5,6\right\}$ and $\left\{1,2,4,5,5,6\right\}$ are not permutations of the set $\left\{1,2,3,4,5,6\right\}$ .

Now suppose you have these objects:

1, 2, 3

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

1 2 3; $\phantom{\rule{0.277778em}{0ex}}$ 1 3 2; $\phantom{\rule{0.277778em}{0ex}}$ 2 1 3; $\phantom{\rule{0.277778em}{0ex}}$ 2 3 1; $\phantom{\rule{0.277778em}{0ex}}$ 3 1 2; $\phantom{\rule{0.277778em}{0ex}}$ 3 2 1. $\phantom{\rule{0.277778em}{0ex}}$

## 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 $r\le n$ ).

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 $\left(n-1\right)$ elements to choose from the remaining set.
3. The third member of the permutation can be filled in $\left(n-2\right)$ 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 $\left(n-\left(r-1\right)\right)=\left(n-r+1\right)$ ways.
5. Summarizing, we find that there is a total of
$n\left(n-1\right)\left(n-2\right)...\left(n-r+1\right)$
different permutations of $r$ objects, taken from a pool of $n$ objects. This number is denoted by $P\left(n,r\right)$ and can be written in factorial notation as:
$P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}.$

For example, if we have a total of 5 elements, the integers $\left\{1,2,3,4,5\right\}$ , 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, $P\left(5,3\right)=5!/7!=60!$ .

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 $\left(n-1\right)$ possible positions for the second object. With the first two placed, there are $\left(n-2\right)$ remaining possible positions for the third object; and so on. There are only two positions to choose from for the penultimate object, and the $nth$ object will occupy the last remaining position.

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

$n\left(n-1\right)\left(n-2\right)...2×1=n!$

ways of constructing an ordered sequence of $n$ objects.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers! By Steve Gibbs By Cath Yu By Maureen Miller By Jessica Collett By John Gabrieli By Mary Cohen By OpenStax By Jugnu Khan By Sheila Lopez By Jonathan Long