In how many ways can the letters of the word ACUMEN be rearranged such that the vowels alwaysappear together?

• The word “rearranged” indicates that this is a permutation problem.
  
  The base set {A, C, U, M, E, N} has no indistinguishable objects.
  
  Repetition is not allowed.
  
  Since the 3 vowels must appear together, treat the three as an inseparable unit. Hence, reduce the base set to{{A, U, E}, C, M, N}.
  
  Now, there are 4 different units in the base set, and they can be arranged in
  4P4= 4!ways.
  
  The unit {A, U, E} can itself be internally arranged in
  3P3= 3! ways.
  
  Hence, by The FundamentalPrinciple of Counting, the total number of ways of arranging the word is 4!x 3!.
• 12 years agoHelpfull: Yes(47) No(7)
• ACUMEN
  AUE will come together as a group which can be arranged in 3!=6 ways.
  AUE, C,M,N can be arranged in 4!=24 ways.
  
  total possible arrangements = 24*6=144 ways
• 12 years agoHelpfull: Yes(22) No(3)
• A,C,U is taken as one letter
  hence there are 4 letters and the 3 vowels can arrange among themselves in 3! ways
  Hence ans is 4!*3!=144
• 12 years agoHelpfull: Yes(6) No(1)
• 4!*3!=144
  there are three constants and three vowels,three vowels always comes together so total number of words are 4.and three vowels can be arranged in 3! ways.
• 12 years agoHelpfull: Yes(3) No(1)
• here we are asked for rearrangement ,it means that it is already arranged once so the answer will be (4!-3!)-1=143
• 10 years agoHelpfull: Yes(2) No(1)
• vowels together rules:
  1)take out vowels from given letter, count the rest and add 1, find facto
  2)find facto for vowels and multiply the result with (1)
  so ans is:
  ACUMEN => CMN(AEU)=> now totally 4,so 4!=>24
  now for vowels AEU=>3!=>6
  24*6=144
• 11 years agoHelpfull: Yes(1) No(0)
• ans is 4!*3!=144
• 10 years agoHelpfull: Yes(1) No(0)
• 4!*3! as we have to assign vowels together so we hv left wid 4 groups..arranging 4 together is 4! nd 3 letters can also b arranged in 3! ways
• 10 years agoHelpfull: Yes(0) No(0)
• vowels are A U E so we unite them and remaining 3 letters and vowels together we have 4 units so we can arrange 4! and vowels can be arranged in 3! ways so we can arrange in 4!*3!= 144 ways
• 8 years agoHelpfull: Yes(0) No(0)