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# Different words are formed using the letters of the word 'INTEGER'. If out of these words, $x$ is number of words in which $I\:and\:N$ are never together and $y$ is no. of words which begin with I and end with R, then $\large\frac{x}{y}=?$

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

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$