How many ways are there to list the letters of the word ALABAMA?

To determine how many ways there are to list (or arrange) the letters of the word "ALABAMA," we need to use the concept of permutations of a multiset, where some elements are repeated.

The word "ALABAMA" has 7 letters in total, and the letters have the following frequencies:
- A appears 4 times.
- L appears 1 time.
- B appears 2 times.
  
The formula for finding the number of distinct permutations of a multiset is:

\[
\text{Number of distinct permutations} = \frac{n!}{k_1! \, k_2! \, \dots \, k_r!}
\]

where:
- \(n\) is the total number of letters (in this case, 7),
- \(k_1, k_2, \dots, k_r\) are the frequencies of the distinct letters.

For "ALABAMA," the calculation is:

\[
\text{Number of ways} = \frac{7!}{4! \, 2! \, 1!}
\]

First, calculate the factorials:
- \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)
- \(4! = 4 \times 3 \times 2 \times 1 = 24\)
- \(2! = 2 \times 1 = 2\)
- \(1! = 1\)

Now, substitute the values into the formula:

\[
\text{Number of ways} = \frac{5040}{24 \times 2 \times 1} = \frac{5040}{48} = 105
\]

So, there are **105 distinct ways** to arrange the letters of "ALABAMA."

Would you like more details or have any questions?

Here are 5 related questions:
1. How many ways are there to arrange the letters of the word "MISSISSIPPI"?
2. What is the formula for calculating permutations of distinct objects?
3. How do permutations differ from combinations?
4. Can you explain why we divide by the factorials of repeated letters?
5. How would the calculation change if there were no repeated letters?

**Tip:** Always check for repeated elements when counting permutations, as this reduces the total number of distinct arrangements.

Learn how to calculate the number of distinct ways to arrange the letters of the word ALABAMA using the formula for permutations of multisets. This detailed explanation breaks down the factorial calculations and explains the concept of repeated elements. Perfect for students in Grades 9-12.

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