Question

How many different words can be formed by using all the letters of the word $ALLAHABAD$?In how many of them vowels occupy the even positions?

Answer 1

There are $9$ letters in the word $ALLAHABAD$ out of which $4$ are $A^{\prime }s,2$ are $L^{\prime }s$ and the rest are all distinct.

So, the requisite number of words$=\frac{9!}{4!2!}=\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 2}=9\times 4\times 7\times 6\times 5=7560$

There are $4$ vowels and all are alike i.e. $4A^{\prime }s$. Also, there are $4$ even places viz $2^{nd},4^{th},6^{th}$ and $8^{th}$. 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 $\left(2L^{\prime }s\right)$ and other distinct, can be arranged in $\frac{5!}{2!}$ ways.