Permutation combination where no two vowels occur together
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i have 5 choose 3 ways of putting 3 vowels around the 4 consonants, which gives 10. The consonants can be permuted 4!=24 ways. The vowels can be permuted 3!=6 ways. You multiply these to get the answer of 1440.
If You want to know the number of arrangements that have exactly two vowels touching, you do something similar. You have 5 choose 2 ways to put the vowel pair and the single vowel around the 4 consonants, which gives 10 also. There are 2!=2 ways to choose which of those vowel spots has the vowel pair. There are 3!=6 ways to permute the vowels, 4!=24 ways to permute the consonants. You multiply these, and get 2880.
If you add these two cases together, you get 4320 ways in which not all 3 vowels are together.
If You want to know the number of arrangements that have exactly two vowels touching, you do something similar. You have 5 choose 2 ways to put the vowel pair and the single vowel around the 4 consonants, which gives 10 also. There are 2!=2 ways to choose which of those vowel spots has the vowel pair. There are 3!=6 ways to permute the vowels, 4!=24 ways to permute the consonants. You multiply these, and get 2880.
If you add these two cases together, you get 4320 ways in which not all 3 vowels are together.
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Answer:
6 (removing one duplicate overlapping of all three together from previous as this will happen twice) = 1440. Ans C. Bunuel wrote: The letters of the word PROMISE are arranged so that no two of the vowels should come together.
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