The letters of word $Z E N I T H$ are written in all possible ways. If all these words are written out as in a dictionary, then find the rank of the word $Z E N I T H$ .

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Solution

Firstly arrange alphabatically
EHINTZ

$\begin{matrix} E H I N T \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 5 \times 5!$ \
$\begin{matrix} \begin{matrix} Z E H Z E I \end{matrix}) 2 \times 3! Z E N H - 1 \times 2! Z E N I H - 1! Z E N I T H - 1! 5 \times 5! + 2 \times 3! + 2! + 1! + 1! \Rightarrow 600 + 12 + 2 + 1 + 1 \Rightarrow 616 \end{matrix}$

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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.

Firstly arrange alphabatically EHINTZEHINT⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠5×5!\ZEHZEI)2×3!ZENH−1×2!ZENIH−1!ZENITH−1!5×5!+2×3!+2!+1!+1!⇒600+12+2+1+1⇒616

The letters of word $Z E N I T H$ are written in all possible ways. If all these words are written out as in a dictionary, then find the rank of the word $Z E N I T H$ .