How many arrangements of the letters in the word GRACIOUS both begin and end w/ a vowel?

1 Answer
Feb 28, 2016



There are 4 options for vowels which may be in the first or last position hence we need to know in how many ways can you choose 2 from an available 4 and arrange them.
This is equivalent to solving the permutation #""^4P_2=`12#.

However, for each of these 12 ways in which 2 vowels can be chosen for the first and last letter, the remaining 6 letters not chosen can be chosen and arranged in #6! = 720# different ways.

Hence, by the multiplication principle, the total number of different ways that the letters can be arranged subject to the given constraints is #12xx720=8640# different ways.