Question

How many different words can be formed by using all the letters of the word ‘ALLAHABAD’?In how many of them both $L$ do not come together?

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$=\frac{9!}{4!2!}=\frac{9\times 8\times 7\times 6\times 5\times 4!}{4!\times 2}=9\times 4\times 7\times 6\times 5=7560$

Considering both $L$ together and treating them as one letter we have $8$ letters out of which $A$ repeats $4$ times and others are distinct. These $8$ letters can be arranged in $\frac{8!}{4!}$ ways.

So, the number of words in which both $L$ come together$=\frac{8!}{4!}=\frac{8\times 7\times 6\times 5\times 4!}{4!}=8\times 7\times 6\times 5=1680$

Hence, the number of words in which both $L$ do not come together

$=$ Total number of words $–$ No. of words in which both $L$ come together

$=7560–1680=5880$.