The number of ways in which four letters of the word MATHEMATICS can be arranged is given by:

Answer

To find the number of ways to arrange four letters from the word "MATHEMATICS", we need to consider 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

This gives us a total of 11 letters. We will analyze different cases based on the selection of letters.

### Step 1: Identify the Cases
We will consider three cases based on the selection of letters:
1. All four letters are identical.
2. Two letters are identical and two are different.
3. All four letters are different.

### Step 2: Case 1 - All letters are identical
In this case, we can only choose from M, A, or T since they are the only letters that appear more than once. However, since we need four letters, this case is not possible.

**Hint:** Check the frequency of letters to see if you can select four identical letters.

### Step 3: Case 2 - Two letters identical and two different
Here we can select two identical letters from M, A, or T, and then choose two different letters from the remaining letters.

- Choose 2 identical letters (M, A, or T): There are 3 choices (M, A, T).
- Choose 2 different letters from the remaining 7 letters (M, A, T, H, E, I, C, S). Since we have already chosen one of M, A, or T, we will have 7 remaining letters minus the chosen identical letter.

The number of ways to choose 2 different letters from the remaining letters is given by \( \binom{7}{2} \).

The arrangement of the letters will be \( \frac{4!}{2!} \) (since we have two identical letters).

So, the total for this case is:
\[ 3 \cdot \binom{7}{2} \cdot \frac{4!}{2!} \]

**Hint:** Remember to use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

### Step 4: Case 3 - All letters different
In this case, we can select any 4 letters from the 8 different letters (M, A, T, H, E, I, C, S).

The number of ways to choose 4 letters from 8 is given by \( \binom{8}{4} \).

The arrangement of these letters will be \( 4! \).

So, the total for this case is:
\[ \binom{8}{4} \cdot 4! \]

**Hint:** Use the factorial to calculate the arrangements of distinct letters.

### Step 5: Combine the Cases
Now we combine the results from both cases to get the total number of arrangements:
\[ \text{Total} = \left(3 \cdot \binom{7}{2} \cdot \frac{4!}{2!}\right) + \left(\binom{8}{4} \cdot 4!\right) \]

### Step 6: Calculate the Values
1. Calculate \( \binom{7}{2} = \frac{7!}{2!(7-2)!} = 21 \).
2. Calculate \( \frac{4!}{2!} = \frac{24}{2} = 12 \).
3. Calculate \( \binom{8}{4} = \frac{8!}{4!(8-4)!} = 70 \).
4. Calculate \( 4! = 24 \).

Now plug these values into the total:
\[ \text{Total} = (3 \cdot 21 \cdot 12) + (70 \cdot 24) \]
\[ = 756 + 1680 = 2436 \]

### Final Answer
The total number of ways to arrange four letters from the word "MATHEMATICS" is **2436**.