Question 1175422
Find the number of ways of arranging all 12 letters of word STRAWBERRIES where
the first and the last letters are vowels.<pre>
To make things easier, put the letters of STRAWBERRIES in alphabetical order:

A,B,E,E,I,R,R,R,T,S,S,W

The vowels are A,E,E,I

Case 1. No E's on either end. A and I are on the ends.  
That's 2!=2 ways they can go on the two ends.
Between them are the distinguishable 10-letter arrangements of 
B,E,E,R,R,R,T,S,S,W or 10!/(2!3!2!) 

That's 2![10!/(2!3!2!)] = 302400 ways for Case 1.

Case 2. Exactly 1 E on one end. That puts A or I on the opposite end.
Choose the end (left or right) to put the E on in 2 ways.
That's 2! ways to place the E
Choose letter A or I to put on the opposite end in 2 ways.
That's 2!(2) ways to put the vowels on the ends, exactly one being an E.
Between the vowels on the ends are the distinguishable 10-letter arrangements
of X,B,E,R,R,R,T,S,S,W, or 10!/(3!2!), where the X represents A or I, the one
not chosen for the opposite end.
or 10!/(3!2!)  

That's (2!)(2)[10!/(3!2!)] = 1209600 ways for Case 2.

Case 3. E's on both ends.  
Between them are the distinguishable 10-letter arrangements of
A,B,I,R,R,R,T,S,S,W or 10!/(3!2!) = 302400 for case 3.

For all three cases, that's 302400+1209600+302400 = 1814400 ways.  

Edwin</pre>