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 solve the problem of finding the number of 4-letter words that can be formed using the letters from the word "IIT JEE", we will analyze the letters and their frequencies, then categorize the arrangements based on different cases.

### 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

So, we have a total of 6 letters with the following counts:
- I appears 2 times
- T appears 1 time
- J appears 1 time
- E appears 2 times

2. **Case 1: All Letters Different**:
We can select 4 different letters from the available letters (I, T, J, E). However, we can only select 3 different letters since we have only 4 unique letters (I, T, J, E). Thus, we can form words using:
- I, T, J, E

The number of arrangements of 4 different letters is given by:
\[ 4! = 24 \]

3. **Case 2: Two Letters Alike and Two Different**:
Here, we can have two cases based on which letter appears twice:
- Case 2a: Two I's and two different letters (T, J, E)
- Case 2b: Two E's and two different letters (I, T, J)

**Case 2a**: Choose 1 from (T, J, E):
- Choose 1 letter from T, J, E: \( \binom{3}{1} = 3 \)
- Arrangements:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]
Total for this case:
\[ 3 \times 12 = 36 \]

**Case 2b**: Choose 1 from (I, T, J):
- Choose 1 letter from I, T, J: \( \binom{3}{1} = 3 \)
- Arrangements:
\[ \frac{4!}{2!} = \frac{24}{2} = 12 \]
Total for this case:
\[ 3 \times 12 = 36 \]

4. **Case 3: Two Letters Alike and Two Letters Alike**:
Here, we can only have:
- Two I's and two E's.

The arrangement is:
\[ \frac{4!}{2! \times 2!} = \frac{24}{4} = 6 \]

5. **Total Count**:
Now, we add all the cases together:
- Case 1: 24 (all different)
- Case 2: 36 (two I's) + 36 (two E's) = 72
- Case 3: 6 (two I's and two E's)

Total:
\[ 24 + 72 + 6 = 102 \]

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

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