How many arrangements of the letters of the word ‘BENGALI’ can be made
(i) If the vowels are never together.
I dont want the solution by negation method. I have seen the solution on :http://www.lofoya.com/Aptitude-Questions-And-Answers/Permutation-and-Combination/intro.htm which mentions 4320 but i could not get this answer by non-negation method.
Best Answer
If the question is asking how many ways can you arrange the letters so that no two vowels are touching, this is how you solve it. You 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.