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How many arrangements can be made out of the letters of the word COMMITTEE, taken all at a time, such that the four vowels do not come together?

Correct Answer: 43200

Explanation:

There are total 9 letters in the word COMMITTEE in which there are 2M's, 2T's, 2E's.


The number of ways in which 9 letters can be arranged =  9 ! 2 ! × 2 ! × 2 !  = 45360


 


There are 4 vowels O,I,E,E in the given word. If the four vowels always come together, taking them as one letter we have to arrange 5 + 1 = 6 letters which include 2Ms and 2Ts and this be done in  6 ! 2 ! × 2 !  = 180 ways.


 


In which of 180 ways, the 4 vowels O,I,E,E remaining together can be arranged in  4 ! 2 !  = 12 ways.


 


The number of ways in which the four vowels always come together = 180 x 12 = 2160.


 


Hence, the required number of ways in which the four vowels do not come together = 45360 - 2160 = 43200


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