We are required to form different words with the help of the word INTEGER. Let $m_1$ be the number of words in which I and N are never together and $m_2$ be the number or words which begin with I and end with R, then $m_1 / m_2$ is

INTEGER

Arrange T, E, G, E, R in $\frac{5!}{2!}$ ways.

There are 6 gaps –T-E-G-E-R-

In the 6 gaps I, N can be arranged in 6P₂ ways

$\therefore {\mathrm{m}}_1=\frac{5!}{2!}\times 6{\mathrm{P}}_2=60\times 30=1800$

${\mathrm{m}}_2=$ No of words begin with I, end with R.

In between I ,R we can arrange N, T, G, E, E in $\frac{5!}{2!}=60$ ways

$\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}=\frac{1800}{60}=30$

option 2