Question 495196
The vowels and consonants, respectively, are AIIIIU and CCDGLRSTT. Assuming two of the same letter are indistinguishable, there are


*[tex \LARGE N = \frac{15!}{4!2!2!}] ways to arrange the letters.


Consider only the vowels. Out of the 6!/4! = 30 distinguishable ways to arrange them, only one is in alphabetical order. For the consonants, out of the 9!/2!2! = 90720 ways to arrange, only one is in alphabetical order, so we divide N by 30 and then by 90720, leaving 5005 ways.