Find the number of words formed containing 4 letters taken from the letters of the word 'INEFFECTIVE'.

Answer

To find the number of 4-letter words that can be formed from the letters of the word "INEFFECTIVE", we will analyze the letters and their frequencies, then consider different cases based on the repetition of letters.

### Step-by-Step Solution:

1. **Identify the Letters and Their Frequencies**:
The word "INEFFECTIVE" consists of the following letters:
- I: 2 times
- N: 1 time
- E: 3 times
- F: 2 times
- C: 1 time
- T: 1 time
- V: 1 time

Total letters = 11.

2. **Identify Distinct Letters**:
The distinct letters are I, N, E, F, C, T, V. This gives us a total of 7 distinct letters.

3. **Consider Different Cases**:
We will consider different cases based on the repetition of letters.

**Case 1: All 4 letters are distinct**
We can choose 4 letters from the 7 distinct letters.
- Number of ways to choose 4 letters: \( \binom{7}{4} \)
- Arrangements of these 4 letters: \( 4! \)

Total for Case 1:
\[ \text{Total}_1 = \binom{7}{4} \times 4! = 35 \times 24 = 840 \]

**Case 2: 2 letters are the same and 2 are distinct**
The letters that can be repeated are I, E, and F. We can choose one of these letters to be repeated.
- Number of ways to choose the letter to repeat: \( \binom{3}{1} \)
- Choose 2 distinct letters from the remaining 6 letters (after choosing the repeated letter): \( \binom{6}{2} \)
- Arrangements: \( \frac{4!}{2!} \)

Total for Case 2:
\[ \text{Total}_2 = \binom{3}{1} \times \binom{6}{2} \times \frac{4!}{2!} = 3 \times 15 \times 12 = 540 \]

**Case 3: 2 letters of one type and 2 letters of another type**
The only possible combination is 2 I's and 2 E's.
- Arrangements: \( \frac{4!}{2! \times 2!} \)

Total for Case 3:
\[ \text{Total}_3 = 1 \times \frac{4!}{2! \times 2!} = 1 \times 6 = 6 \]

**Case 4: 3 letters are the same and 1 is distinct**
The only letter that can be repeated 3 times is E.
- Choose 1 distinct letter from the remaining 6 letters: \( \binom{6}{1} \)
- Arrangements: \( \frac{4!}{3!} \)

Total for Case 4:
\[ \text{Total}_4 = 1 \times \binom{6}{1} \times \frac{4!}{3!} = 1 \times 6 \times 4 = 24 \]

4. **Calculate Total Words**:
Now, we sum up all the cases:
\[ \text{Total} = \text{Total}_1 + \text{Total}_2 + \text{Total}_3 + \text{Total}_4 = 840 + 540 + 6 + 24 = 1410 \]

### Final Answer:
The total number of 4-letter words that can be formed from the letters of the word "INEFFECTIVE" is **1410**.