$\begin{array}{1 1} 42 \\ 36 \\ 30 \\ \frac{1}{30} \end{array}$

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No. of words using letters of the word 'INTEGER' $= \large\frac{7!}{2}$

No. of words in which I and N are together $=\large\frac{6!}{2}$

$\therefore$ No. of words in which I and N are never together $=\large\frac{7!}{2}$$-\large\frac{6!}{2}$

$\Rightarrow\:x=\large\frac{6.6!}{2}$

No. of words which begin with I and end with R=$\large\frac{5!}{2}$

$\Rightarrow\:y=\large\frac{5!}{2}$

$\Rightarrow\:\large\frac{x}{y}$$=36$

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