In how many ways can the letters of the word COMBINE be arranged so that vowels and consonants do not come together.
The word COMBINE has 3 vowels and 4 consonants. If the letters have to be arranged such that the vowels and consonants are not together, the consonants should be placed at positions 1, 3, 5 and 7 and the vowels should be placed at positions 2, 4 and 6.
There are 4 consonants that can be placed at position 1, 3 are left to place at position 3, 2 are left to place at position 5 and the last is placed at position 7. This gives the number of arrangements as 4*3*2*1 = 24. Similarly, at position 2 there are 3 vowels that can be placed, at position 4 the remaining 2 vowels can be placed and at position 6 the last vowel can be placed. This gives the number of possible arrangements as 3*2*1 = 6. In all there are 24*6 = 144 ways of arranging the letters.
The required number of arrangements that satisfy the condition that vowels and consonants are not together is 144.