since the vowels are in alphabetical order E should always follow A

 STARTING FROM LEFT
 
 A AT 1ST POSITION E CAN BE PLACED AT REMAINING 5 PLACES =5C1*4!
 
 A AT 2ND POSITION E CAN BE PLACED AT REMAINING 4 PLACES =4C1*4!

 

 A AT 3RD POSITION E CAN BE PLACED AT REMAINING 3 PLACES =3C1*4!
 
 A AT 4TH POSITION E CAN BE PLACED AT REMAINING 2 PLACES =2C1*4!

 

 A AT 5TH POSITION E CAN BE PLACED AT REMAINING 1 PLACES =1C1*4!

 

 A  AT 6TH POSITION E CANNOT BE PLACED AT REMAINING PLACES OTHERWISE E WOULD COME BEFORE A
 
THEREFORE TOTAL NUMBER OF WAYS= 4!*(5C1+4C1+3C1+2C1+1C1)=360 WAYS