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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? |
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Answer» 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. 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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