In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?  Choose the correct choice.
A) 620  B) 640  C) 690  D) 710  E) 720
The word 'OPTICAL' has 7 letters. It has the vowels 'O','I','A' in it and these 3 vowels should always come together. Hence these three vowels can be grouped and considered as a single letter. That is, PTCL(OIA).   Hence we can assume total letters as 5 and all these letters are different.   Number of ways to arrange these letters   =   5   !   =   5   ×   4   ×   3   ×   2   ×   1   =   120   =5!=5×4×3×2×1=120   All the 3 vowels (OIA) are different   Number of ways to arrange these vowels among themselves   =   3   !   =   3   ×   2   ×   1   =   6   =3!=3×2×1=6   Hence, required number of ways   =   120   ×   6   =   720   720)
<result>E</result>
In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?