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 and I and end with R, then `m_1/m_2`

Options

A. 42
B. 30
C. 6
D. `1/30`

Answer

To solve the problem, we need to find the values of \( m_1 \) and \( m_2 \) based on the arrangements of the letters in the word "INTEGER".

### Step 1: Calculate the total number of arrangements of the letters in "INTEGER".

The word "INTEGER" consists of 7 letters where 'E' appears twice. The formula for the total arrangements of letters, accounting for repetitions, is given by:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

where \( n \) is the total number of letters and \( p_i \) are the frequencies of the repeated letters.

For "INTEGER":
- Total letters \( n = 7 \)
- The letter 'E' is repeated \( 2 \) times.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{7!}{2!} \]

Calculating this:

\[ 7! = 5040 \quad \text{and} \quad 2! = 2 \]

\[ \text{Total arrangements} = \frac{5040}{2} = 2520 \]

### Step 2: Calculate \( m_1 \) (the number of arrangements where I and N are never together).

To find \( m_1 \), we first calculate the arrangements where I and N are together. We treat "IN" as a single block.

Now, we have the blocks: "IN", T, E, G, E, R. This gives us 6 blocks in total (considering "IN" as one block).

The arrangements of these 6 blocks (where 'E' is still repeated) are:

\[ \text{Arrangements with I and N together} = \frac{6!}{2!} \]

Calculating this:

\[ 6! = 720 \]

\[ \text{Arrangements with I and N together} = \frac{720}{2} = 360 \]

Now, we can find \( m_1 \):

\[ m_1 = \text{Total arrangements} - \text{Arrangements with I and N together} \]

\[ m_1 = 2520 - 360 = 2160 \]

### Step 3: Calculate \( m_2 \) (the number of arrangements that start with I and end with R).

For \( m_2 \), we fix I at the start and R at the end. The letters left to arrange are T, E, G, E, N (5 letters).

The arrangements of these 5 letters (with 'E' repeated) are:

\[ m_2 = \frac{5!}{2!} \]

Calculating this:

\[ 5! = 120 \]

\[ m_2 = \frac{120}{2} = 60 \]

### Step 4: Calculate \( \frac{m_1}{m_2} \).

Now we can find the ratio of \( m_1 \) to \( m_2 \):

\[ \frac{m_1}{m_2} = \frac{2160}{60} = 36 \]

### Final Answer:

\[ \frac{m_1}{m_2} = 36 \]