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To find the number of distinguishable arrangements of the letters in the word "NONILLION," we need to consider the total number of letters and the frequency of each distinct letter.

### Steps to Solve:
1. **Count the Total Letters**: The word "NONILLION" has 9 letters.
2. **Identify Repeated Letters**:
   - 'N' appears 3 times.
   - 'O' appears 2 times.
   - 'I' appears 2 times.
   - 'L' appears 2 times.

### Formula:
The formula for the number of distinguishable permutations of letters in a word is:

\[
\text{Number of arrangements} = \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}
\]

Where:
- \( n \) is the total number of letters.
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the repeated letters.

### Calculation:
\[
\text{Number of arrangements} = \frac{9!}{3! \times 2! \times 2! \times 2!}
\]

Now, let's compute this:

- \( 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)
- \( 2! = 2 \times 1 = 2 \)

\[
\text{Number of arrangements} = \frac{362880}{6 \times 2 \times 2 \times 2} = \frac{362880}{48} = 7560
\]

So, there are **7,560** distinguishable arrangements of the letters in the word "NONILLION."

Would you like more details on any part of this solution? Here are some related questions that might interest you:

1. How would the formula change if the word had no repeated letters?
2. What would be the number of arrangements for a word with different frequencies of repeated letters?
3. How does the factorial operation apply in combinatorial problems?
4. What is the significance of arranging repeated elements in permutations?
5. Can this approach be used for non-alphabetical characters (e.g., digits or symbols)?

**Tip**: Always break down problems involving permutations into their basic elements—consider the total and the repeats separately.

Find the number of distinguishable arrangements of the letters of the word 'NONILLION.'

Learn how to calculate the number of distinguishable arrangements of the letters in the word NONILLION. This combinatorial problem involves using the permutation formula to account for repeated letters, such as N appearing 3 times. Suitable for high school students.

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