In how many ways the word CINEMA can be rearranged such that the order of vowel does not changed.

• Ans: 120
  The order of vowel does not change only means that the order of vowels which is i,e,a should not be changed.Taking different cases of occurrences,
  In the order e after i and a after e,
  When i in pos1,
  e in pos2, no. of possible arrangements = 4*3!
  e in pos 3, no. of possible arrangements =3*3!
  e in pos 4, no. of possible arrangements =2*3!
  e in pos 5, no. of possible arrangements =1*3!
  Total = (4+3+2+1)*3!=60
  When i in pos2,
  e in pos 3, no. of possible arrangements =3*3!
  e in pos 4, no. of possible arrangements =2*3!
  …………..and so on.
  Total = (3+2+1)*3!=36
  When i in pos3,
  e in pos 4, no. of possible arrangements =2*3!
  and so on.
  Total = (2+1)*3!=18
  Similarly for i in pos4, total = 1*3!=6
  There can’t be cases of i in pos 5 or e in pos6 because i,e,a order need to be maintained.
  So total no. of possible arrangements = 60+36+18+6 = 120.
  
• 11 years agoHelpfull: Yes(35) No(15)
• since the 3 vowels i.e i,e and a will be fixed at their place. so rest 3 places will be filled by 3 consonants. first place has 3 choices among the 3 consonants, then next has 2 choices and the last has only one choice. so total way will be 3!=6ways
• 11 years agoHelpfull: Yes(26) No(22)
• ans-4!
  vowel-IEA(consider as a one letter)
  consonant-CNM(three letter)
  since, the order of vowel remain same(not place)hence the ans will be (3+1)!=4!
• 11 years agoHelpfull: Yes(18) No(10)
• only 6 ways....eg.CINEMA,CIMENA,NIMECA,NICEMA,MICENA,MINECA
• 11 years agoHelpfull: Yes(9) No(13)
• simple logic is let us assume 3 positions are fixed in every case these could be any where in possible 6 positions such that 3 vowels come at these 3 positions in the desired order. now we are left with 3 non vowel terms these are to be arranged in any 3 out of 6 places...so applying permutation(bcz for arrangements permutation is used and for selection combination is used) ans will be 6p3=120.
• 10 years agoHelpfull: Yes(8) No(2)
• I,E,A can be placed as a 1 letter so left place will be 4 so i,e,a, in same order can be placed in four ways and consonents now can arrange as 3! ways so answer will be 24 ways.
• 11 years agoHelpfull: Yes(6) No(6)
• 3 vowels can arrange in 3! ways and consonants can arrange in 3! ways
  
• 11 years agoHelpfull: Yes(5) No(14)
• i don't whether this is correct. but they asked only order should not change. but the even place and odd places can change right.so
  case1: vowels in even order position _i_e_a ...1st plce in 3 ways,2nd in 2 ways and 3rd in 1 way so total--6 ways
  case2:vowels in odd place order i_e_a_ ... ist place in 3 ways, 2nd in 2ways and 3rd in 1 way so total 6 ways
  altogether 12 ways. ( i guess)
• 11 years agoHelpfull: Yes(4) No(11)
• c i n e m a
  constant is. c,n,m
  vowel is. i e a
  so
  3!*3!
  36
• 11 years agoHelpfull: Yes(4) No(7)
• consider all 3 vowels(i,e,a) as one unit and there wl be 4 elments rearranged in
  4! ways...
  
• 11 years agoHelpfull: Yes(4) No(8)
• 4!.cin+(iea)=4!
  
• 11 years agoHelpfull: Yes(4) No(8)
• ans will be 36
  because we will put vowels and consonents in following way
  C V C V C V
  consonents can be arranged in 3! i.e 6 ways
  order of vowels not changed but means vowels should come at 2nd 4rth and 6th places so they can be arranged in 3! i.e 6 ways
  so total 6*6=36 ways
• 10 years agoHelpfull: Yes(3) No(0)
• 3! will be the answer
• 11 years agoHelpfull: Yes(2) No(7)
• The three vowels must be kept in the same order. Now there are a total of three consonants to arrange. This can be done in 3! = 6 ways.
• 11 years agoHelpfull: Yes(2) No(6)
• firstly consonants are CNM and vowels are IEA
  then take CNM as 3!and IEA as 1!
  combinely then letters to be arranged as CNM(IEA)=4!
  VOWELS COME TOGETHER =4!*3!=24*6=144
  total no of words formed using the letters CINEMA is =6!=720
  VOWELS DOESN'T COME TOGETHER=720-144=576
  576 is the answer
• 10 years agoHelpfull: Yes(2) No(2)
• 4!
  as iea is considered 1 and cnm as 3 so they can be placed ad 4!.....iea is in same order...not in same place
• 10 years agoHelpfull: Yes(2) No(0)
• @ANN MARY JOLLY
  I DONT THINK THIS IS THE WAY TO SOLVE THIS QUESTION.
  WHEN THE QUESTION ALREADY SAID THAT WE CANT CHANGE THE POSITIONS OF THE VOWELS,SO WHAT IS THE POINT OF TESTING THE CASES.I THINK THE RIGHT ANSWER IS 3!=6WAYS.AND AS WE CANT CHANGE THE POSITIONS OF THE VOWELS.SO WE KEEP IT INTACT IN THERE POSITIONS
• 10 years agoHelpfull: Yes(2) No(0)
• Ans = 120
  
  "I E A" should be in same sequence..
  
  a short alternative to find it is:-
  
  since there are 3! arrangements of "I E A":-
  I E A like in 'cinema' etc
  I A E like in 'nimeca' etc
  E A I like in 'eacmin' etc
  E I A like in 'eiacmn' etc
  A I E like in 'caiemn' etc
  A E I like in 'acemin' etc
  
  so, in all these six sequence of VOWELS can be found in any of the word formed by arrangements of 6 letters of "CINEMA"
  
  Out of these six sequence only 1 is acceptable i.e. "I E A"
  
  so, divide total possible cases by 6 to get the answer..
  
  => (6!)/6 = 120
• 10 years agoHelpfull: Yes(2) No(1)
• first arrange i,a,e in 3!ways
  next there will be 4 gaps,we arrange c,n,m in 4p3 ways
  there answer is 3!*4p3=140ways
  
• 11 years agoHelpfull: Yes(1) No(4)
• This is best thought of in two steps.
  
  Step one is to choose the places that the vowels go. Here we are picking three places out of six, and the order that we do this is not important. This is a combination and there are a total of C(6 ,3) = 20 ways to perform this step.
  
  The remaining five letters may be arranged in 3! = 6 ways. This gives a total of 20 x 6 = 120 arrangements.
• 10 years agoHelpfull: Yes(1) No(0)
• this is right...
  not order change (place nhi) so IEA consider 1 letter
  nd rest of CNM 3 letters
  
  _ _ _ _ =4! Ans
• 10 years agoHelpfull: Yes(1) No(0)
• suppose I will be placed in first place then remaining two vowels can be arranged in 4*4 ways then total word will be (4*4*3!)
  in the same manner for placing I in second,third.fourth are (3*3*3!),(2*2*3!),(1*1*3!) respectively...then finally
  3!(4*4+3*3+2*2+1*1)=180 ways
• 11 years agoHelpfull: Yes(0) No(8)
• 18 3 changing places 6 letters 6*3
• 11 years agoHelpfull: Yes(0) No(4)
• vowels order is i,e,a .there are 2 cases
  1.i,e,a comes together (4!=24 ways)
  2. i,e,a does not come together(5ways)
  total answer 29 ways
• 11 years agoHelpfull: Yes(0) No(6)
• i think its 3! because all the vowels are fixed.. ryt.. nd remaining 3 letters are arranged in 3! ways
• 11 years agoHelpfull: Yes(0) No(3)
• ans is:- 11 ways.
  
  here the order of vowels does not changed.
  Case 1:-
  (-I-E-A)
  for this we have 3! ways= 6 ways
  Case 2:-
  (I-E-M-)
  again for this we are having 3! ways= 6 ways.
  So total is 12 ways.
  
  So there are total 11 ways of rearranging
• 10 years agoHelpfull: Yes(0) No(0)
• 6!/3!=120
  
• 10 years agoHelpfull: Yes(0) No(0)
• total position = 6.
  C,N,M can be arrnged in 6p3 way=120 way..
  I,E,A can b arrngd in 1 way..
  so total arrngmnts = 6p3*1=120 way..
• 9 years agoHelpfull: Yes(0) No(0)
• In the word CINEMA we have
  
  3 vowels (I,E,A)...They can be arranged in 3! ways=3!=3*2=6.
  
  3 consonants(C,N,M)...They can be arranged in 3! ways=3!=3*2=6
  
  Total no of ways they can arrange= 3!*3!
  = 6*6
  =36 ways
• 7 years agoHelpfull: Yes(0) No(0)
• selection of 3 positions from 6 different positions is 6p3
  and the remaining position letters can be arranged in 3!
  so total ways= 6p3*3!=120
• 7 years agoHelpfull: Yes(0) No(0)