**Permutations: Introduction**

Permutations are a mathematical concept used to determine the number of possible arrangements or orders in which a set of items can be arranged. In the context of the given question, we need to determine the number of arrangements that can be made using the letters of the word "permutation".

**Understanding the Word**

The word "permutation" consists of 11 letters: p, e, r, m, u, t, a, t, i, o, n. To find the number of arrangements, we need to consider the total number of letters and their repetitions.

**Counting the Repetitions**

In the word "permutation", we have the following repetitions:
- The letter "t" appears twice.
- The letter "a" appears twice.

To calculate the number of arrangements, we can use the concept of permutations with repetitions.

**Permutations with Repetitions**

When dealing with permutations with repetitions, we use the formula:

\[ P = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdot ... \cdot n_k!} \]

Where:
- P represents the total number of arrangements.
- n represents the total number of items (in this case, the total number of letters).
- n1, n2, n3, ..., nk represent the number of repetitions for each letter.

**Calculating the Number of Arrangements**

Using the formula for permutations with repetitions, we can now calculate the number of arrangements for the word "permutation":

\[ P = \frac{11!}{2! \cdot 2!} \]

Breaking down the calculation:

11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

2! = 2 × 1

The calculation becomes:

\[ P = \frac{11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1}{(2 × 1) × (2 × 1)} \]

Simplifying further:

\[ P = \frac{11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3}{2 × 1} \]

\[ P = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 \]

Calculating this expression gives us the total number of arrangements that can be made using the letters of the word "permutation".

11×10×9×8×7×6×5×4×3