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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.

Applications

The binomial theorem

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

( x + y ) n = k = 0 n n k x k y n - k

Whenever n is a positive integer, the numbers

n k = n ! k ! ( n - k ) !

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

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

( x + y ) 2 = x 2 + 2 x y + y 2 ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4

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 Khayyám 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 = 200 000

  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 = 8 000 000

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

    Probability = 200 000 8 000 000 = 1 40 = 0 , 025

Show that

n ! ( n - 1 ) ! = n
  1. Method 1: Expand the factorial notation.

    n ! ( n - 1 ) ! = n × ( n - 1 ) × ( n - 2 ) × . . . × 2 × 1 ( n - 1 ) × ( n - 2 ) × . . . × 2 × 1

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

    So n ! ( n - 1 ) ! = n

  2. Method 2: We know that P ( n , r ) = n ! ( n - r ) ! 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 n ! ( n - 1 ) ! = 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 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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
what's the program
Jordan
?
Jordan
what chemical
Jordan
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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