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Permutation and combination

Three pictures are to be hung in line on a wall. Indicating the different pictures by A, B and C, one order in which they can be hung is A, B, C and another is A, C, B.

Each of these arrangements is called a permutation of the three pictures (and there are further possible permutations).

i.e. a permutation is an ordered arrangement of a number of items.

Suppose, however, that seven pictures are available for hanging and only three of them can be displayed. This time a choice has first to be made. Representing the seven pictures by A, B, C, D,E,F and G, one possible choice of the three pictures for display is A, B, and C. Regardless of the order in which they are then hung this group of three is just one choice and is called combination. Thus A, B, C or A, C, B or B, A, C or B, C, A or C, A, B or C, B, A are six different permutations but only one combination .

i.e. a combination is an unordered selection of a number of items from a given set.

In general, n! represents the number nx(n-1)x(n-2)x…x2x1, i.e. n! means the product of all the integers from 1 to n inclusive.

Exercises :

  1. In each of the following problems determine , without working out the answer, whether you are asked to find a number of permutations, or a number of combinations.
    • How many arrangements of the letters A, B, C are there ?
    • A team of six members is chosen from a group of eight. How many different teams can be selected ?
    • A person can take eight records to a desert island, chosen from his own collection of one hundred records. How many different sets of records could he choose ?
    • One red die and one green die are rolled (each number one to six). In how many ways can a total score of six be obtained ?
  2. a) How many different combinations of six letters can be chosen from the letters A, B, C, D, E, F, G, H, if each letter is chosen only once ?

b) In how many ways can the eight letters be divided into two groups of six and two letters ?

  1. A team of four children is to be selected from a class of twenty children , to compete in a quiz game. In how many ways can the team be chosen if :
    • any four children can be chosen
    • the four chosen must include the oldest in the class ?
  2. A shop stocks ten different varieties of packet soup. In how many ways can a shopper buy three packets of soup if :
    • each packet is a different variety
    • two packet are the same variety
  3. a) In how many ways can ten different books be divided into two groups of six and four books ?

b) In how many different hands of five cards can be dealt from a suit of thirteen cards?

  1. A large box of biscuits contains nine different varieties. In how many ways can four biscuits be chosen if :
    • all four are different
    • two are the same and the others different
    • two each of two varieties are selected
    • there are the same and the fourth is different
    • all four the same
  2. Find how many distinct numbers greater than 5000 and divisible by 3 can be formed from the digits 3,4,5,6 and 0, each digit being used at most once in any number.
  3. A certain test consists of seven questions, to each of which a candidate must give one of three possible answers. According to the answer that he chooses, the candidate must score 1, 2, or 3 marks for each of the seven questions. In how many different ways can a candidate score exactly 18 marks in the test?
  4. A tennis club is to select a team of three pairs, each pair consisting of a man and a woman, for a match. The team is to be chosen from 7 men and 5 women. In how many different ways can the three pairs be selected?
  5. How many four digit odd numbers can be made from the set {5, 7, 8, 9}, no integer being used more than once?
  6. How many numbers greater than 4000 can be made from the set {1,3,5,7}, if each integer can be used only once?
  7. How many arrangements can be made of three letters chosen from PEAT if the first letter is a vowel and each arrangement contains three different letters?
  8. How many three digit numbers can be made from the set of integers {1,2,3,4,5,6,7,8,9} if:

(a) the three digits are all different,

(b) the three digits are all the same,

(c) the number is greater than 600,

(d) all three digits are the same and the number is odd?

  1. Three boxes each contain three identical balls. The first box has red balls in it, the second blue balls and the third green balls. In how many ways can three balls be arranged in a row if:

(a) the balls are of different colours,

(b) all three balls are of the same colour?

  1. n red counters and m green counters are to be placed in a straight line. Find the number of different arrangements of the colours. A town has n streets running from south to north and m streets running from west to east. A man wishes to go from the extreme south-west intersection to the extreme north-east intersection, always moving either north or east along one of the streets. Find the number of different routes he can take.

Questions & Answers

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
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
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
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Tổ hợp và hoán vị. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10796/1.1
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