MADAM ,CIVIC

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The total number of ways of arranging ORANGE = 6! The total number of ways o,a and e can be arranged = 3! The total number of groups when the vowels are one group and the rest are individuals= 4! First of all we have to find all vowels occur together There are 6 different letters in the word ORANGE, in which there are 3 vowels, namely, O, A and E. Since the vowels have to occur together, we can for the time being, assume them as a single object (OAE). This single object together with 3 remaining letters (objects) will be counted as 4 objects. Then we count permutations of these 4 objects taken all at a time. This number would be 4P4 = 4!. Corresponding to each of these permutations, we shall have 3! permutations of the three vowels O, A, E taken all at a time . Hence, by the multiplication principle the required number of permutations = 4 ! × 3 ! = 144. Now we need In how many different ways can the letters of the word 'ORANGE' be arranged so that the three vowels never come together so If we have to count those permutations in which all vowels are never together, we first have to find all possible arrangements of 6 letters taken all at a time,which can be done in 6! ways. Then, we have to subtract from this number, the number of permutations in which the vowels are always together. The Required number =6!-4!3! =576

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