In how many different ways can the letters of the word 'MATHEMATICS' be arranged such that the vowels must always come together?
Select the correct option.
A) 9800	B) 100020	C) 120960	D) 140020	E) 150000
Explanation:   The word 'MATHEMATICS' has 11 letters. It has the vowels 'A','E','A','I' in it and these 4 vowels must always come together. Hence these 4 vowels can be grouped and considered as a single letter. That is, MTHMTCS(AEAI).   Hence we can assume total letters as 8. But in these 8 letters, 'M' occurs 2 times, 'T' occurs 2 times but rest of the letters are different.   Hence,number of ways to arrange these letters   =8!(2!)(2!)=8!(2!)(2!) =8Ã—7Ã—6Ã—5Ã—4Ã—3Ã—2Ã—1(2Ã—1)(2Ã—1)=10080=8Ã—7Ã—6Ã—5Ã—4Ã—3Ã—2Ã—1(2Ã—1)(2Ã—1)=10080   In the 4 vowels (AEAI), 'A' occurs 2 times and rest of the vowels are different.   Number of ways to arrange these vowels among themselves =4!2!=4Ã—3Ã—2Ã—12Ã—1=12=4!2!=4Ã—3Ã—2Ã—12Ã—1=12   Hence, required number of ways   =10080Ã—12=120960   ANSWER IS 120960
<result>C</result>
In how many different ways can the letters of the word 'MATHEMATICS' be arranged such that the vowels must always come together?