 Hence find number of arrangements of the letters in the word
 'MATHEMATICA'

The word "MATHEMATICA" consists of 11 letters. Some of these letters are repeated:

- M appears 2 times.
- A appears 3 times.
- T appears 2 times.
- H, E, I, and C each appear once.

To find the number of distinct arrangements of these letters, we use the formula for permutations of a multiset:

\[
\text{Number of arrangements} = \frac{n!}{k_1! \times k_2! \times \dots \times k_r!}
\]

Where:
- \( n \) is the total number of letters (11 in this case),
- \( k_1, k_2, \dots, k_r \) are the factorials of the frequencies of the repeated letters.

For "MATHEMATICA", the formula becomes:

\[
\frac{11!}{2! \times 3! \times 2! \times 1! \times 1! \times 1! \times 1!}
\]

Let's calculate this step by step:

\[
11! = 39916800
\]
\[
2! = 2 \quad \text{and} \quad 3! = 6
\]
Thus,

\[
\frac{39916800}{2! \times 3! \times 2!} = \frac{39916800}{2 \times 6 \times 2} = \frac{39916800}{24} = 1663200
\]

So, the number of distinct arrangements of the letters in "MATHEMATICA" is **1,663,200**.

Would you like any further details or clarifications?

Here are 5 related questions you might find interesting:
1. How many arrangements are possible if one letter is omitted from "MATHEMATICA"?
2. What if all letters in "MATHEMATICA" were distinct—how many arrangements would there be?
3. How would the number of arrangements change if the word had one more repeated letter?
4. Can we generalize this formula for any word with repeated letters?
5. How many arrangements of "MATHEMATICA" start with a specific letter?

**Tip**: Always look for repeated elements in permutations to simplify calculations!

Hence find number of arrangements of the letters in the word 'MATHEMATICA'

Explore how to calculate the number of distinct arrangements of the letters in 'MATHEMATICA'. This problem uses combinatorics and factorials to solve a multiset permutation problem. The final result is 1,663,200 distinct arrangements. Suitable for students in Grades 9-12.

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