The letters of the word ‘MIRROR’ are permuted in all possible ways and the words thus obtained are arranged in alphabetical order as in a dictionary, and ranked. Then the Rank of the given word is

Alphabetical order : $IMORRR$ 

To find the rank of the word $MIRROR$

The number words :

          $$\begin{gathered} starting with I = \left(\frac{5!}{3!}\right) \\ \text{'}\text{'} \text{'}\text{'} MIO =\left(\frac{3!}{3!}\right) \\ \text{'}\text{'} \text{'}\text{'} MIRO = \left(\frac{2!}{2!}\right) \end{gathered}$$

First word starting with MIRR is MIRROR

$Rank=1\left(\frac{5!}{3!}\right)+1\left(\frac{3!}{3!}\right)+1\left(\frac{2!}{2!}\right)+1=23$