# 4.6 Counting rules - university of calgary - base content - v2015

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This module covers Permutations and Combinations

## Some counting rules

Instead of writing out all possible outcomes for an experiment, we can quickly find the total number of possible outcomes. We have already discussed that when tossing a coin twice, there are two trials which result in 4 different outcomes (S = {HH, HT, TH, TT}). We can use a tree diagram to represent this in the figure below.

For tossing the coin 3 times, we get 8 possible outcomes S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }. The tree diagram is displayed below.

From the tree diagram we can see that the number of outcomes increases by a factor of 2 with each trial.

A die is rolled twice.

1. How many possible outcomes are there?
2. Write out the sample space.

1. There are 6 possible outcomes for the first role and 6 possible outcomes for the second role.

$6\times 6$ possible outcomes

2. The figure below is a visual aid of the possible outcomes. For example, you could get “1” on the first roll and “2” on the second roll. This is a different outcome fromgetting “2” on the first roll and “1” on the second roll.

## The multiplicative rule

A quick way to get the total number of possible outcomes without writing out the sample space or creating a visual aid is to multiply the number of possible outcomes for each trial.

When tossing a coin twice, there are two possible outcomes for each trial (H,T) regardless of whether the coin is weighted or not. If we multiply the two outcomes for the first trial and the two outcomes for the second trial we get $2\times 2=4$ possible outcomes. S = { HH, HT, TH, TT }.

When tossing a coin three times, there are two possible outcomes for each trial (H,T) and we end up with 8 possible outcomes. From the tree diagram we saw that we get S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }.

$2\times 2\times 2=8$

We can extend this concept to tossing a coin 4 times where there are 16 possible outcomes. We will leave it up to you to write out the sample space.

$2\times 2\times 2\times 2=16$

## Try it

You are going to toss a coin and roll a die.

1. Calculate the number of possible outcomes if you toss the coin once and roll the die once.
2. Calculate the number of possible outcomes if you toss the coin twice and roll the die once.
3. Calculate the number of possible outcomes if you toss the coin twice and roll the die twice.
1. There are 2 possible outcomes for tossing the die and 6 possible outcomes for rolling the die. $2\times 6=12$
2. $2\times 2\times 6=24$
3. $2\times 2\times 6\times 6=72$

Suppose we wish to arrange four pictures in a row along a wall. How many different outcomes are possible?

There are four pictures that can be selected for the first position on the wall. 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 $4\times 3=12$ possible arrangements of pictures for the first two positions on the wall. If we continue with this procedure, we now only have two pictures to choose from for the third position and the last picture goes in the last position. As a result, there are 24 possible arrangements of the pictures in a row along a wall.

#### Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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