The letters of the word ‘FATHER’ are taken and permuted in all possible ways. The words thus formed are arranged as in a dictionary and ranked. Then the rank of the given word is

Alphabetical order : AEFHRT

To find the rank of the word FATHER 

The number of words :

        $$\begin{gathered} starting with A = 5! \\ \text{'}\text{'} \text{'}\text{'} E = 5! \\ \text{'}\text{'} \text{'}\text{'} FAE = 3! \\ \text{'}\text{'} \text{'}\text{'} FAH = 3! \\ \text{'}\text{'} \text{'}\text{'} FAR = 3! \\ \text{'}\text{'} \text{'}\text{'} FATE = 2! \end{gathered}$$

The next word in dictionary order is FATHER

$\frac{\bcancel{A}\bcancel{E}\bcancel{F}\bcancel{H}\bcancel{R}\bcancel{T}}{\bcancel{F}\bcancel{A}\bcancel{T}\bcancel{H}\bcancel{E}\bcancel{R}}$  $=\left( 2x(5!)+ 3x(3!) + 1x(2!)+1\right)=261$ 

Then the rank of the given word is 261