Question

We have to form different words with the letters of the word $INTEGER$. Let $m_1$ be the number of words in which $I\&N$ are never together and $m_2$ be the number of words which begin with $I$ and end with $R$, then $\frac{m_1}{m_2}$is equal to?

• A. $30$
• B. $60$
• C. $90$
• D. $180$

Answer 1

The correct option is A

$30$

Explanation for correct option:

Given, $INTEGER$,

Total letters $=7$

Total number of words can be formed $\frac{7!}{2!}$ as $E$ is repeated twice.

Number of words form when $I\&N$ comes together $\frac{6!}{2!}$as $E$ is repeated twice.

$I\&N$ itself arrange $2!$

Therefore, Number of words form when $I\&N$ comes together $\frac{6!.2}{2!}$

Number of words, $I\&N$ never together, $m_1=\frac{7!}{2!}-\frac{6!.2}{2!}=\frac{6!.5}{2!}$

Number of words, when $I$ and end with $R$,$m_2=\frac{5!}{2!}$ as $E$ is repeated twice

Therefore, $\begin{array}{rcl}\frac{m_1}{m_2}& =& \frac{6!.5/2!}{5!/2!}\\ & =& 30\end{array}$

Hence correct option is $\mathbf{\left(}\mathit{A}\mathbf{\right)}$