How many different words can be formed by using all the letters of the word
Question:

How many different words can be formed by using all the letters of the word ‘ALLAHABAD’?

Solution:

Given: We have 9 letters

To Find: Number of words formed with Letter of the word ‘ALLAHABAD.’

The formula used: The number of permutations of $n$ objects, where $p_{1}$ objects are of one kind, $p_{2}$ are of the second kind, …, $p_{k}$ is of a $k^{\text {th }}$ kind and the rest if any, are of a

different kind is $=\frac{n !}{p_{1} ! p_{2} ! \ldots \ldots \ldots \ldots p_{k} !}$

‘ALLAHABAD’ consist of 9 letters out of which we have 4 A’s and 2 L’s.

Using the above formula

Where,

$n=9$

$p_{1}=4$

$p_{2}=2$

$\Rightarrow \frac{9 !}{4 ! 2 !}=7560$

7560 different words can be formed by using all the letters of the word ‘ALLAHABAD.’