Question

In how many ways Can the word COMMITTEE be arranged​

Answer 1

43200

Explanation:

There are total 9 letters in the word COMMITTEE, had they been all unique letters then total permutations = 9! = 362880

Now, we have 2M's, 2T's and 2E's, so the total permutations of COMMITTEE will be (9!)/(2! * 2! * 2!) = 362880/8 = 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, so the permutations = (6!)/(2! * 2!) = 720/4 = 180

In the 180 arrangements, the 4 vowels O,I,E,E remain together and can be arranged in (4!)/(2!) ways = 24/2 = 12

So, 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

Sorry if it's wrong.

Answer 2

Answer is 43200

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
No of possible arrangement = 45360 - 2160 = 43200