The no.of arrangements that can be formed out of ' LOGARITHM 'so that all the vowels not come to gether.
There are 9 different alphabets used in the word LOGARITHM.
These alphabets can be arranged in 9! ways to form words.
There are three vowels (O, A and I) in the word.
Consider these vowels as a single letter. This single letter with the remaining 6 consonants can be arranged in (6 + 1)! = 7! ways.
These three vowels can be arranged in 3! ways.
So, the total number of ways of arranging the letters to form words in which vowels come together is 7! × 3!.
Hence, total number of arrangements in which all vowels will not come together = 9! - 7! × 3!.
Consider that V1, V2, V3 be the three vowels and C1, C2, C3, C4, C5 and C6 be the six consonants.
Inorder not keep any vowels together, the positions of consonants and vowels can be given as follows. Here shading portion represents that C3 may come any of the shading boxes.
So, there are a total of 5 + 4 + 3 + 2 + 1 + 4 + 3 + 2 + 1 + 3 + 2 + 1 + 2 + 1 + 1 = 35 positions of V3 on the basis of positions of V1 and V2..
V1, V2 and V3 may interchange their places3! ways and the 6 consonants can be interchanged in 6! ways.
Therefore, no vowels come together in 35 × 3! × 6! ways.