Find out the number of ways in which the word ARRANGE can be arranged such that there is at least one vowel between two consonants.

ARRANGE

There are 3 vowels - A,A,E

and 4 consonants - R,R,N,G



The consonants can be arranged in : 4! / 2! ways

Now, in order to arrange in such a manner that there is atleast one vowel between any 2 consonants, in this case we need a vowel between every 2 consonants.

There are 3 such positions.

Therefore, the vowels can be placed in: 3!/2! ways



So the total ways are:

(4! / 2!) * (3! / 2!)

= 36 ways.  (edited calculation mistake)