Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the latter M appears at the fourth position in any such word is :

Options

A. `(1)/(66)`
B. `(1)/(11)`
C. `(1)/(9)`
D. `(2)/(11)`

Answer

To find the probability that the letter 'M' appears at the fourth position in any arrangement of the letters of the word "EXAMINATION", we can follow these steps:

### Step 1: Count the total number of letters in "EXAMINATION"
The word "EXAMINATION" consists of 11 letters.

### Step 2: Identify the frequency of each letter
The letters in "EXAMINATION" and their frequencies are:
- E: 1
- X: 1
- A: 1
- M: 1
- I: 2
- N: 2
- T: 1
- O: 1

### Step 3: Calculate the total arrangements of the letters
The total number of arrangements of the letters in "EXAMINATION" can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{11!}{2! \times 2! \times 2!} \]

Where \( 2! \) accounts for the two 'I's, two 'N's, and two 'A's.

### Step 4: Calculate the favorable outcomes where 'M' is at the fourth position
If we fix 'M' at the fourth position, we are left with 10 letters to arrange (E, X, A, I, I, N, N, T, O). The arrangements of these 10 letters can be calculated as:

\[ \text{Favorable arrangements} = \frac{10!}{2! \times 2!} \]

Where \( 2! \) accounts for the two 'I's and two 'N's.

### Step 5: Calculate the probability
The probability that 'M' is at the fourth position is given by the ratio of the number of favorable arrangements to the total arrangements:

\[ \text{Probability} = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{\frac{10!}{2! \times 2!}}{\frac{11!}{2! \times 2! \times 2!}} \]

This simplifies to:

\[ \text{Probability} = \frac{10! \times 2!}{11!} = \frac{1}{11} \]

### Final Answer
The probability that the letter 'M' appears at the fourth position in any arrangement of the letters of the word "EXAMINATION" is:

\[ \frac{1}{11} \]