How many different words can be formed by using all the letters o( the word 'ALLAHABAD' such that vowels  occupy the even positions, is

There are 4 vowels and all are alike i.e. 4A's. Also, there are 4 even places viz 2nd, 4th, 6th and 8th. So, these 4 even places can be occupied by 4 vowels in $\frac{4!}{4!}=1$ way. Now, we are left with 5 places in which 5 letters, of which two are alike (2 L's) and other distinct, can be arranged in $\frac{5!}{2!}$ ways.

Hence, the total number of words in which vowels occupy the even places $=\frac{5!}{2!}\times \frac{4!}{4!}=\frac{5!}{2!}=60$