Question

in how many different ways can t e letters of the word "LEADING" be arranged in such a way that vowels always come together?

Answer 1

The word 'LEADING' has 7 different letters.  When the vowels EAI are always together, they can be supposed to form one letter.  Then, we have to arrange the letters LNDG (EAI).  Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.  The vowels (EAI) can be arranged among themselves in 3! = 6 ways.  Required number of ways = (120 x 6) = 720.  The word 'LEADING' can be arranged 720 different ways in such a way that vowels always come together. 

Answer 2

We have 3 vowels. The number of ways we can arrange them so they are next to each other is 3!=6. Now we have to find the number of ways we can arrange these 3 vowels with the remaining letters. As the vowels have to come together, we can treat them as one letter. Therefore we have 5 letter altogether. The number of ways we can arrange the vowels with the remaining letters is 5!=120. 

6*120=720