Question

We are to form different words with the letters of the word ‘INTEGER’. Let $m_{1}$ be the number of words in which I and Nare never together, and $m_{2}$ be the number of words which begin with I and end with R. Then $m_{1}/m_{2}$ is equal to

• A. 30
• B. 60
• C. 90
• D. 180

Answer 1

Answer: (A)

30

Answer 2

We have 5 letters other than ‘I’ and ‘N’ of which two are identical (E's). We can arrange these letters in a line in $\frac{5!}{2!}$ ways. In any such arrangement ‘I’ and ‘N’ can be placed in 6 available gaps in $6P_{2}$ ways, so required number = $\frac{5!}{2!}^{6}P_{2} = m_{1}$.

Now, if word start with I and end with R then the remaining letters are 5. So, total number of ways = $\frac{5!}{2!} = m_{2}$.

$\therefore$ $\frac{m_{1}}{m_{2}} = \frac{5!}{2!}.\frac{6!}{4!}.\frac{2!}{5!} = 30$.`