In how many different ways can the letters of the word MATHEMATICS be arranged ?
MATHEMATICS शब्द के अक्षरों को कितने अलग-अलग तरीकों से व्यवस्थित किया जा सकता है?

Options

A. `(10!)/(2)`
B. `(11!)/(2)`
C. `(11!)/(4)`
D. `(11!)/(8)`

Answer

To find the number of different ways to arrange the letters of the word "MATHEMATICS", we can follow these steps:

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

### Step 2: Identify the repeated letters
In the word "MATHEMATICS":
- The letter 'M' appears 2 times.
- The letter 'A' appears 2 times.
- The letter 'T' appears 2 times.
- The letters 'H', 'E', 'I', 'C', and 'S' each appear 1 time.

### Step 3: Use the formula for permutations of multiset
The formula to calculate the number of arrangements of letters when there are repeated letters 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 letters,
- \( p_1, p_2, p_3, \ldots \) are the frequencies of the repeated letters.

In our case:
- \( n = 11 \) (total letters)
- \( p_1 = 2 \) (for 'M')
- \( p_2 = 2 \) (for 'A')
- \( p_3 = 2 \) (for 'T')

### Step 4: Substitute the values into the formula
Using the values we have:

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

### Step 5: Calculate the factorials
Now we can calculate the factorials:
- \( 11! = 39916800 \)
- \( 2! = 2 \)

Thus, we have:

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

### Final Answer
The number of different ways to arrange the letters of the word "MATHEMATICS" is **4,989,600**.

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