To find the number of all possible arrangements of the letters of the word "MATHEMATICS" taken four at a time, we will break down the problem into different cases based on the repetition of letters.
### Step 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
- C: 1
- I: 1
- S: 1
Total letters = 11 (M, A, T, H, E, M, A, T, I, C, S)
### Step 2: Consider different cases for selection of letters
We will consider three cases based on how many letters are repeated.
#### Case 1: All four letters are different
We can select 4 different letters from the 8 unique letters (M, A, T, H, E, C, I, S).
- Number of ways to choose 4 letters from 8 = \( \binom{8}{4} \)
- Each selection can be arranged in \( 4! \) ways.
Calculating:
\[
\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
\]
\[
4! = 24
\]
Total arrangements for Case 1:
\[
70 \times 24 = 1680
\]
#### Case 2: Two letters are the same, and two letters are different
Here, we can choose one letter that appears twice (M, A, or T) and two other different letters from the remaining unique letters.
- Choose the letter that appears twice: 3 choices (M, A, T)
- Choose 2 different letters from the remaining 7 unique letters.
Calculating:
\[
\text{Choose 2 from 7} = \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21
\]
The arrangement of these letters (e.g., MMXY) can be done in:
\[
\frac{4!}{2!} = 12 \text{ ways}
\]
Total arrangements for Case 2:
\[
3 \times 21 \times 12 = 756
\]
#### Case 3: Two pairs of letters are the same
In this case, we can choose two letters that appear twice (M, A, T).
- Choose 2 letters from {M, A, T}: \( \binom{3}{2} = 3 \)
- The arrangement of these letters (e.g., MMAA) can be done in:
\[
\frac{4!}{2!2!} = 6 \text{ ways}
\]
Total arrangements for Case 3:
\[
3 \times 6 = 18
\]
### Step 3: Sum all cases
Now we add the total arrangements from all cases:
\[
\text{Total arrangements} = 1680 + 756 + 18 = 2454
\]
### Final Answer
The total number of all possible arrangements of the letters of the word "MATHEMATICS" taken four at a time is **2454**.
