Question

In word 'ALLAHABAD' how many of them vowels occupy the even positions?

Answer 1

There are 4 vowels and all are alike i.e. A'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 which5 letters, of which two are alike (2L'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
$=\frac{5!}{2!}=\frac{5!}{2!}=160$