The number of different arrangements that can be made by using all the letters in the word MATHEMATICS is……

Answer

To find the number of different arrangements that can be made using all the letters in the word "MATHEMATICS", we follow these steps:

### Step 1: Count the total number of letters
The word "MATHEMATICS" consists of 11 letters.

### Step 2: Identify the frequency of each letter
In the word "MATHEMATICS", the frequency of each letter is as follows:
- M: 2 times
- A: 2 times
- T: 2 times
- H: 1 time
- E: 1 time
- I: 1 time
- C: 1 time
- S: 1 time

### Step 3: Use the formula for permutations of multiset
The formula for the number of different arrangements (permutations) of letters in a word where some letters are repeated is given by:

\[ \text{Number of arrangements} = \frac{N!}{n_1! \times n_2! \times n_3! \times \ldots} \]

where:
- \(N\) is the total number of letters,
- \(n_1, n_2, n_3, \ldots\) are the frequencies of the repeated letters.

### Step 4: Substitute the values into the formula
Here, \(N = 11\) (total letters), and the repeated letters are:
- M: 2 times
- A: 2 times
- T: 2 times

Thus, the formula becomes:

\[ \text{Number of arrangements} = \frac{11!}{2! \times 2! \times 2!} \]

### Step 5: Calculate the factorials
Now, we calculate the factorials:

- \(11! = 39916800\)
- \(2! = 2\)

### Step 6: Substitute the factorial values into the equation
Now we can substitute these values into our formula:

\[ \text{Number of arrangements} = \frac{39916800}{2 \times 2 \times 2} = \frac{39916800}{8} = 4989600 \]

### Final Answer
The number of different arrangements that can be made by using all the letters in the word "MATHEMATICS" is **4,989,600**.

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