Number of words of 4 letters that can be formed with the letters of the word IIT JEE, is

Options

A. 42
B. 82
C. 102
D. 142

Explanation

To find the number of 4-letter words that can be formed using the letters of the word "IIT JEE", we will analyze the problem by considering different cases based on the repetition of letters. The letters in "IIT JEE" are I, I, T, J, E, E.

### Step-by-step Solution:

1. **Identify the Letters and Their Frequencies**:
- The letters in "IIT JEE" are: I (2), T (1), J (1), E (2).
- We have a total of 6 letters: I, I, T, J, E, E.

2. **Case 1: All Letters Different**:
- We can select 4 different letters from the available letters. However, since we only have 4 unique letters (I, T, J, E), we can only use these.
- The number of ways to choose 4 letters from 4 is given by \( \binom{4}{4} = 1 \).
- The arrangement of these 4 different letters can be done in \( 4! = 24 \) ways.
- **Total for Case 1**: \( 1 \times 24 = 24 \).

3. **Case 2: Two Letters Alike and Two Different**:
- Here, we can have either II with two other different letters or EE with two other different letters.
- **Sub-case 2.1: Using II**:
- We can choose 2 different letters from T, J, E. The possible selections are:
- T and J
- T and E
- J and E
- This gives us \( \binom{3}{2} = 3 \) ways to choose the other two letters.
- The arrangement of these letters (II, T, J) or (II, T, E) or (II, J, E) can be done in \( \frac{4!}{2!} = 12 \) ways for each selection.
- **Total for Sub-case 2.1**: \( 3 \times 12 = 36 \).

- **Sub-case 2.2: Using EE**:
- We can choose 2 different letters from I, T, J. The possible selections are:
- I and T
- I and J
- T and J
- This gives us \( \binom{3}{2} = 3 \) ways to choose the other two letters.
- The arrangement of these letters (EE, I, T) or (EE, I, J) or (EE, T, J) can also be done in \( \frac{4!}{2!} = 12 \) ways for each selection.
- **Total for Sub-case 2.2**: \( 3 \times 12 = 36 \).

- **Total for Case 2**: \( 36 + 36 = 72 \).

4. **Case 3: Two Letters Alike of One Kind and Two Alike of Another Kind**:
- The only combination possible is II and EE.
- The arrangement of these letters (II, EE) can be done in \( \frac{4!}{2! \times 2!} = 6 \) ways.
- **Total for Case 3**: \( 6 \).

5. **Final Calculation**:
- Adding all the cases together:
- Case 1: 24
- Case 2: 72
- Case 3: 6
- Total number of 4-letter words = \( 24 + 72 + 6 = 102 \).

### Conclusion:
The total number of 4-letter words that can be formed with the letters of the word "IIT JEE" is **102**.