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The number of ways of arranging n unlike objects in a line is n! .

The number of ways of arranging in a line is n objects , of which p are alike , is.

The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are different is (n-1)! .

The number of ways of arranging n unlike objects in a ring when clockwise and anticlockwise arrangements are the same, is.

e.g. The representatives of five countries attend a conference. In how many ways can they be seated at a round table ?

As they are seated at a round table, there is no first or last place to consider. What matters in this arrangement is where each one sits relative to the others, as no one seat is special.

EDCBA12345DCBAE12345CBAED12345BAEDC12345AEDCB12345Numbering the chairs 1 to 5, we can say that there are 5 ways of selecting the occupant of the first seat . 4 ways of selecting the occupant of the second seat … and so on. Thus there are 5x4x3x2x1 ways of arranging the representatives in the five seats . But this number includes the arrangements shown below :

i.e. for any one arrangement in the five seats , the representatives can be moved clockwise five times and still have the same people to the left and to the right of them . Therefore the 5x4x3x2x1 ways of arranging the five representatives in numbered seats is five times the number of ways of arranging them round a circular table , so there areways, or 4x3x2x1 ways = (5-1)! in which the representatives can be seated at the round table.

e.g. In how many ways can five beads, chosen from eight different beads be threaded on to a ring ?

EDCBA12345DCBAE12345CBAED12345BAEDC12345AEDCB12345The number of ways of arranging five beads taken from eight different beads , in five numbered places is 8x7x6x5x4. Thus the number of ways of arranging five (from eight) beads in a circle isas :

But a ring can be turned over,


and these have been counted as two separate arrangements. So the number of circular arrangements is twice the number of arrangements on a ring. Therefore the number of ways of threading five beads, from eight different beads , on a ring is.


  1. In how many ways can eight people be seated at a round table?
  2. In how many ways can eight cows be placed in a circular milking parlour?


  1. 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 ?
  2. In how many ways can the eight letters be divided into two groups of six and two letters?

3. In how many ways can five different books be arranged on a shelf ?

4. How many two digit numbers can be made from the set {2,3,4,5,6,7,8,9}, each number containing two different digits ?

5. In how many ways can six different shrubs be planted in a row?

6. In how many of the possible permutations of the letters of the word ADDING are the two D's :

  1. together
  2. separate

7. How many permutations of the letters of the word DEFEATED are there in which the E's are separated from each other ?

8. A box contains seven snooker balls , three of which are red , two black , one white and one green. In how many ways can three balls be chosen ?

9. Four books are taken from a shelf of eighteen books , of which six are paperback, and twelve are hardback. In how many of the possible combinations of four books is at least one a paperback ?

10. Find how many numbers between 10 and 300 can be made from the digits 1, 2, 3, if :

  1. each digit may be used only once,
  2. each digit may be used more than once ?

11. How many combinations of three letters taken from the letters A, A, B, B, C, C, D are there ?


  1. Find the number of permutations of the letters of the word MATHEMATICS
  2. Find the number of permutations of four letters from the word MATHEMATICS
  3. How many of the permutations contain two pairs of letters that are the same ?

13. Find the number of ways in which twelve children can be divided into two groups of six if two particular boys must be in different groups ?

14. A box contains ten bricks , identical except for colour. Three bricks are red , two are white, two are yellow , two are blue and one is black. In how many ways can three distinguishable bricks be

  1. taken from the box
  2. arranged in a row ?
  3. In how many of the arrangements in a row of all ten bricks are the three red bricks separated from each other ,
  4. In how many of the arrangements in a row of all ten bricks are just two of the red bricks nect to each other ?

15. How many even numbers less than 500 can be made from the integers 1,2,3,4,5, each integer being used only once ?

16. In how many ways can three letters from the word GREEN be arranged in a row if at least one of the letters is E ?

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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