To find the number of 4-letter words that can be formed from the letters of the word "MATHEMATICS," we will analyze the letters and their frequencies, and then categorize the arrangements based on the repetition of letters.
### Step 1: Count the letters in "MATHEMATICS"
The word "MATHEMATICS" consists of the following letters:
- M: 2
- A: 2
- T: 2
- H: 1
- E: 1
- I: 1
- C: 1
- S: 1
### Step 2: Identify cases based on letter repetition
We will consider three cases based on the repetition of letters in the 4-letter words:
**Case 1:** Two letters are alike of one kind and two letters are alike of another kind.
**Case 2:** Two letters are alike of one kind and two letters are different.
**Case 3:** All four letters are different.
### Step 3: Calculate for each case
#### Case 1: Two alike of one kind and two alike of another kind
- We can choose 2 letters from M, A, T (which appear twice).
- The number of ways to choose 2 letters from {M, A, T} is \( \binom{3}{2} = 3 \).
- The arrangement of these letters (e.g., MM, AA) is given by:
\[
\frac{4!}{2! \times 2!} = 6
\]
- Therefore, the total for this case is:
\[
3 \times 6 = 18
\]
#### Case 2: Two alike of one kind and two different
- Choose 1 letter from {M, A, T} (which appear twice): \( \binom{3}{1} = 3 \).
- Choose 2 different letters from {H, E, I, C, S} (5 letters): \( \binom{5}{2} = 10 \).
- The arrangement is:
\[
\frac{4!}{2!} = 12
\]
- Therefore, the total for this case is:
\[
3 \times 10 \times 12 = 360
\]
#### Case 3: All four letters different
- We can choose 4 letters from the 8 different letters {M, A, T, H, E, I, C, S}.
- The number of ways to choose 4 letters is \( \binom{8}{4} = 70 \).
- The arrangement of these letters is:
\[
4! = 24
\]
- Therefore, the total for this case is:
\[
70 \times 24 = 1680
\]
### Step 4: Sum up all cases
Now, we sum the totals from all three cases:
\[
\text{Total} = 18 + 360 + 1680 = 2058
\]
### Final Answer
The total number of 4-letter words that can be formed from the letters of the word "MATHEMATICS" is **2058**.
