Find out how many words can be formed out of the letters of the word DAUGHTER such that

(i)  The vowels are all time together.

(ii)  The vowels occupy even places. 

 Ans:  In the phrase DAUGHTER, there are three vowels: A, E and U. Number of letters in the word is 8. 

(i)  While vowels are all time together, the number of ways these letters can be arranged (5 +1)! * 3!. The 5 consonants and one group of vowels. Within each such type of arrangement, the three vowels can be arranged in 3! Ways. The needed answer is 4320.

(ii) While vowels occupy even position, after that the three vowels can be placed at 4 even positions in 3*2*1 ways. The three even positions can be chosen in ⁴C₃ ways. And now the 5 consonants can be arranged in 5! Ways. Hence the required number is = 3! * ⁴C₃ *5! = 2880.