# 4.6 Counting rules - university of calgary - base content - v2015  (Page 2/6)

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$4\times 3\times 2\times 1=24$

This can be represented by 4! (read “four factorial”).

$4!=4\times 3\times 2\times 1=24$

## Factorials

The exclamation symbol after a natural number indicates to multiple a series of descending natural numbers from n to 1 .

$n!=n\times n-1\times n-2\times ...\times 1$
0!=1

Suppose that we have five members on a committee.

1. If there are two positions on the committee of president and vice president, how many different ways could the positions be filled?
2. If there are five positions on the committee, how many different ways could you fill the five positions?

1. There are 20 different ways to fill the position of president and vice president. There are five members to choose from for the first position and if we choose a member to be president, we now have four members to choose from for the next position.

$5\times 4=20$

2. $5!$ = $5\times 4\times 3\times 2\times 1=120$

## Using ti-83,83+,84,84+ calculator

• Enter 5.
• Press MATH.
• Arrow across to PRB.
• Press ENTER.
• Arrow down the list to 4:!
• Press ENTER.

20 students have volunteered to be class representatives (reps).

1. If the teacher randomly selects two students to be class representatives, how many groups of two can be formed?
2. If the teacher randomly selects four students to be class representatives, how many groups of four can be formed?
3. If the teacher selects all 20 students to be class representatives and each one is assigned to a group of students, how many different ways can theclass representatives be assigned?

1. $20\times 19=380$
2. $20\times 19\times 18\times 17=116280$
3. $20!$

Let’s look again at the example of hanging pictures along a wall. Suppose you decide that you are only going to hang two of the pictures in a row. We saw that if we choose one picture to hang first, we are now left with three choices of pictures for the next position on the wall. Using the multiplicative rule, there are 43 = 12 possible arrangements of pictures for the first two positions on the wall.

There are four different pictures and we are selecting only two and arranging them in a specific order. This is known as a permutation .

## Permutation

Permutation is the arrangements of r elements in a different order chosen from n distinct available items. Below are different ways that a permutation can be represented.Insert paragraph text here.

${P}_{r}^{n}={}_{n}{P}_{r}=\frac{n!}{(n-r)!}=n(n-1)n(n-2)\mathrm{...}(n-r+1)$

For the picture example, there are four pictures ( n = 4) and we are selecting two pictures ( r = 2) and arranging them in a specific order. Using the multiplication rule, we have seen that the answer is 12. Using permutation, we can see that we get the same result.

${P}_{2}^{4}=$ ${}_{4}{P}_{2}=$ $\frac{4!}{(4-2)!}=$ $\frac{4!}{2!}=$ $\frac{4\times 3\times 2\times 1}{2\times 1}=$ $4\times 3=12$

There are 4! different ways of arranging the four pictures on the wall. We divide by the number of ways of arranging the items that are not selected because we only care about the arrangement of the items selected.

There are 4! different ways of arranging the four pictures on the wall. We divide by the number of ways of arranging the items that are not selected because we only care about the arrangement of the items selected.

For example, let’s label the pictures A , B , C and D . If we write out the sample space for arranging 4 pictures along a wall, we get the sample space

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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