Find how many arrangements can be made with the letters of the word ' MATHEMATICS'?

Answer

To find the number of arrangements that can be made with the letters of the word "MATHEMATICS", we will follow these steps:

### Step 1: Count the total number of letters
The word "MATHEMATICS" consists of the following letters:
- M: 2
- A: 2
- T: 2
- H: 1
- E: 1
- I: 1
- C: 1
- S: 1

Now, let's count the total number of letters:
- Total letters = 2 (M) + 2 (A) + 2 (T) + 1 (H) + 1 (E) + 1 (I) + 1 (C) + 1 (S) = 11 letters.

### Step 2: Identify the repetitions
From the counts above, we can see that:
- M appears 2 times
- A appears 2 times
- T appears 2 times
- H, E, I, C, S each appear 1 time

### Step 3: Use the formula for arrangements
The formula for the number of arrangements of n items where there are repetitions is given by:

\[ \text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times p_3! \times \ldots} \]

Where:
- \(n\) is the total number of items,
- \(p_1, p_2, p_3, \ldots\) are the frequencies of the repeated items.

In our case:
- \(n = 11\)
- The repeating letters are M, A, and T, each appearing 2 times.

Thus, we can substitute into the formula:

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

### Step 4: Calculate the factorials
Now we will calculate \(11!\) and \(2!\):

\[ 11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39916800 \]
\[ 2! = 2 \times 1 = 2 \]

### Step 5: Substitute and simplify
Now we can substitute these values back into the formula:

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

### Final Answer
Thus, the total number of arrangements that can be made with the letters of the word "MATHEMATICS" is **4,989,600**.

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