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Question

The letters of word $$ZENITH$$ are written in all possible ways. If all these words are written out as in a dictionary, then find the rank of the word $$ZENITH$$.


Solution

Firstly arrange alphabatically 
 EHINTZ

$$\left. \begin{array}{l} E \\ H \\ I \\ N \\ T \end{array} \right) 5\times 5!$$\
$$ \begin{array}{l} \left. \begin{array}{l} ZEH \\ ZEI \end{array} \right) \, \, \, 2\times 3! \\ ZENH\, -\, 1\times 2! \\ ZENIH\, \, -\, 1! \\ ZENITH\, -\, \, 1! \\ 5\times 5!\, \, +2\times 3!\, +2!+1!+1! \\ \Rightarrow 600\, \, \, \, \, +12+2+1+1 \\ \Rightarrow 616 \end{array}$$

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