To solve the problem of finding the number of 4-letter words that can be formed from the letters of the word "MATHEMATICS", we need to consider the repetitions of letters in the word and categorize our approach based on different cases of letter combinations.
### Step-by-Step Solution:
1. **Identify the Letters and Their Frequencies**:
The word "MATHEMATICS" consists of the following letters:
- M: 2
- A: 2
- T: 2
- H: 1
- E: 1
- I: 1
- C: 1
- S: 1
Total unique letters = 8 (M, A, T, H, E, I, C, S).
2. **Case 1: All Letters Different**:
We can choose 4 different letters from the 8 unique letters. The number of ways to choose 4 letters from 8 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.
\[
\text{Ways to choose 4 letters} = \binom{8}{4}
\]
After choosing 4 letters, we can arrange them in \( 4! \) ways.
\[
\text{Total for this case} = \binom{8}{4} \times 4! = 70 \times 24 = 1680
\]
3. **Case 2: 2 Same and 2 Different**:
Here, we can choose one letter that appears twice (either M, A, or T) and then choose 2 different letters from the remaining 7 letters.
- Choose 1 letter to be repeated: \( \binom{3}{1} \) (choosing M, A, or T).
- Choose 2 different letters from the remaining 7 letters: \( \binom{7}{2} \).
The arrangement of these letters (2 same and 2 different) can be done in \( \frac{4!}{2!} \) ways.
\[
\text{Total for this case} = \binom{3}{1} \times \binom{7}{2} \times \frac{4!}{2!} = 3 \times 21 \times 12 = 756
\]
4. **Case 3: 2 Same and 2 Same**:
In this case, we can choose two letters that appear twice (M, A, T). We need to choose 2 out of these 3 letters.
- Choose 2 letters from M, A, T: \( \binom{3}{2} \).
- The arrangement of these letters (2 of one kind and 2 of another) can be done in \( \frac{4!}{2! \times 2!} \) ways.
\[
\text{Total for this case} = \binom{3}{2} \times \frac{4!}{2! \times 2!} = 3 \times 6 = 18
\]
5. **Total Count**:
To find the total number of 4-letter words, we sum the results from all cases.
\[
\text{Total} = 1680 + 756 + 18 = 2454
\]
Thus, the total number of 4-letter words that can be formed from the letters of the word "MATHEMATICS" is **2454**.
