SOLUTION: What is the number of distinguishable arrangements that can be made from the word KITCHEN, if the vowels must stay together?

Since the 2 vowels of KITCHEN must stay together, 
we either have to put these 6 things in a row

(K), (IE), (T), (C), (H), (N)

or these 6 things in a row:

(K), (EI), (T), (C), (H), (N)

There are 6! ways to arrange either in a row,

so the answer is 2*6! = 2(720) 1440 ways

Edwin