10.2 Permutations and applications  (Page 2/2)

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Permutation with repetition

When order matters and an object can be chosen more than once then the number of

permutations is:

${n}^{r}$

where $n$ is the number of objects from which you can choose and $r$ is the number to be chosen.

For example, if you have the letters A, B, C, and D and you wish to discover the number of ways of arranging them in three letter patterns (trigrams) you find that there are ${4}^{3}$ or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can choose any of the four letters. Multiplying them together gives the total.

The binomial theorem

In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads

${\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right){x}^{k}{y}^{n-k}$

Whenever $n$ is a positive integer, the numbers

$\left(\genfrac{}{}{0pt}{}{n}{k}\right)=\frac{n!}{k!\left(n-k\right)!}$

are the binomial coefficients (the coefficients in front of the powers).

For example, here are the cases n = 2, n = 3 and n = 4:

$\begin{array}{c}\hfill {\left(x+y\right)}^{2}={x}^{2}+\mathbf{2}xy+{y}^{2}\\ \hfill {\left(x+y\right)}^{3}={x}^{3}+\mathbf{3}{x}^{2}y+\mathbf{3}x{y}^{2}+{y}^{3}\\ \hfill {\left(x+y\right)}^{4}={x}^{4}+\mathbf{4}{x}^{3}y+\mathbf{6}{x}^{2}{y}^{2}+\mathbf{4}x{y}^{3}+{y}^{4}\end{array}$

The coefficients form a triangle, where each number is the sum of the two numbers above it:

This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar KhayyÃ¡m in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.

The number plate on a car consists of any 3 letters of the alphabet (excluding the vowels and 'Q'), followed by any 3 digits (0 to 9). For a car chosen at random, what is the probability that the number plate starts with a 'Y' and ends with an odd digit?

1. The number plate starts with a 'Y', so there is only 1 choice for the first letter, and ends with an even digit, so there are 5 choices for the last digit (1, 3, 5, 7, 9).

2. Use the counting principle. For each of the other letters, there are 20 possible choices (26 in the alphabet, minus 5 vowels and 'Q') and 10 possible choices for each of the other digits.

Number of events = $1×20×20×10×10×5=200\phantom{\rule{0.277778em}{0ex}}000$

3. Use the counting principle. This time, the first letter and last digit can be anything.

Total number of choices = $20×20×20×10×10×10=8\phantom{\rule{0.277778em}{0ex}}000\phantom{\rule{0.277778em}{0ex}}000$

4. The probability is the number of events we are counting, divided by the total number of choices.

Probability = $\frac{200\phantom{\rule{0.277778em}{0ex}}000}{8\phantom{\rule{0.277778em}{0ex}}000\phantom{\rule{0.277778em}{0ex}}000}=\frac{1}{40}=0,025$

Show that

$\frac{n!}{\left(n-1\right)!}=n$
1. Method 1: Expand the factorial notation.

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

Cancelling the common factor of $\left(n-1\right)×\left(n-2\right)×...×2×1$ on the top and bottom leaves $n$ .

So $\frac{n!}{\left(n-1\right)!}=n$

2. Method 2: We know that $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$ is the number of permutations of $r$ objects, taken from a pool of $n$ objects. In this case, $r=1$ . To choose 1 object from $n$ objects, there are $n$ choices.

So $\frac{n!}{\left(n-1\right)!}=n$

Exercises

1. Tshepo and Sally go to a restaurant, where the menu is:
 Starter Main Course Dessert Chicken wings Beef burger Chocolate ice cream Mushroom soup Chicken burger Strawberry ice cream Greek salad Chicken curry Apple crumble Lamb curry Chocolate mousse Vegetable lasagne
1. How many different combinations (of starter, main course, and dessert) can Tshepo have?
2. Sally doesn't like chicken. How many different combinations can she have?
2. Four coins are thrown, and the outcomes recorded. How many different ways are there of getting three heads? First write out the possibilities, and then use the formula for combinations.
3. The answers in a multiple choice test can be A, B, C, D, or E. In a test of 12 questions, how many different ways are there of answering the test?
4. A girl has 4 dresses, 2 necklaces, and 3 handbags.
1. How many different choices of outfit (dress, necklace and handbag) does she have?
2. She now buys two pairs of shoes. How many choices of outfit (dress, necklace, handbag and shoes) does she now have?
5. In a soccer tournament of 9 teams, every team plays every other team.
1. How many matches are there in the tournament?
2. If there are 5 boys' teams and 4 girls' teams, what is the probability that the first match will be played between 2 girls' teams?
6. The letters of the word 'BLUE' are rearranged randomly. How many new words (a word is any combination of letters) can be made?
7. The letters of the word 'CHEMISTRY' are arranged randomly to form a new word. What is the probability that the word will start and end with a vowel?
8. There are 2 History classes, 5 Accounting classes, and 4 Mathematics classes at school. Luke wants to do all three subjects. How many possible combinations of classes are there?
9. A school netball team has 8 members. How many ways are there to choose a captain, vice-captain, and reserve?
10. A class has 15 boys and 10 girls. A debating team of 4 boys and 6 girls must be chosen. How many ways can this be done?
11. A secret pin number is 3 characters long, and can use any digit (0 to 9) or any letter of the alphabet. Repeated characters are allowed. How many possible combinations are there?

Questions & Answers

what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
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
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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