We are asked to find the number of arrangements of the letters in the word MATHEMATICS.

1. Problem Description

We are asked to find the number of arrangements of the letters in the word MATHEMATICS.

2. Solution Steps

The word MATHEMATICS has 11 letters. We count the occurrences of each letter:

M appears 2 times.

A appears 2 times.

T appears 2 times.

H appears 1 time.

E appears 1 time.

I appears 1 time.

C appears 1 time.

S appears 1 time.

The formula for the number of permutations of $n$ objects where there are $n_1$ of one kind, $n_2$ of another kind, ..., and $n_k$ of the $k$-th kind is:

$\frac{n!}{n_1! n_2! ... n_k!}$

In this case, $n = 11$, $n_1 = 2$ (for M), $n_2 = 2$ (for A), $n_3 = 2$ (for T), $n_4 = 1$ (for H), $n_5 = 1$ (for E), $n_6 = 1$ (for I), $n_7 = 1$ (for C), $n_8 = 1$ (for S).

The number of arrangements is:

$\frac{11!}{2! 2! 2! 1! 1! 1! 1! 1!} = \frac{11!}{2! 2! 2!} = \frac{11!}{8}$

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

$\frac{11!}{8} = \frac{39916800}{8} = 4989600$

The calculation from the image says $3 \times 4! = 3 \times 24 = 72$, which seems unrelated. Also the values for the number of repetitions for M, A and T are given.

The number of arrangements of the word MATHEMATICS is $\frac{11!}{2!2!2!} = \frac{11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot 2 \cdot 1 \cdot 2 \cdot 1} = 11 \cdot 10 \cdot 9 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 1 = 4989600$.

3. Final Answer

4989600