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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.

Tree diagram representing toss of coin twice

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.

Tree diagram representing three tosses of coin

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 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.
    Visual aid for rolls of two dice
    sample space for two dice

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 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 2 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 2 2 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 6 12
  2. 2 2 6 24
  3. 2 2 6 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 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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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Source:  OpenStax, Introduction to statistics i - stat 213 - university of calgary - ver2015revb. OpenStax CNX. Oct 21, 2015 Download for free at http://legacy.cnx.org/content/col11874/1.3
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