In how many ways can the letters in the word 'Alaska' be arranged?



The word "Alaska" consists of 6 letters, where some letters are repeated. Specifically, the letter 'A' appears 3 times, and the other letters ('L', 'S', 'K') appear once each.

To find the number of ways to arrange these letters, we use the formula for permutations of multiset:

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

Where:
- \( n \) is the total number of letters (6 in this case),
- \( n_1, n_2, \dots \) are the factorials of the frequencies of the repeated letters.

For "Alaska":
- Total letters: \( n = 6 \),
- The letter 'A' appears 3 times, so \( n_1 = 3 \),
- The letters 'L', 'S', 'K' each appear once, so their factorials are \( n_2 = 1, n_3 = 1, n_4 = 1 \).

Thus, the number of ways to arrange the letters is:

\[
\frac{6!}{3! \cdot 1! \cdot 1! \cdot 1!} = \frac{720}{6} = 120
\]

Therefore, the letters in "Alaska" can be arranged in **120 ways**.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How many ways can the letters of the word "banana" be arranged?
2. What if all letters in the word "Alaska" were distinct, how many arrangements would be possible?
3. How would the solution change if there were 4 A’s in "Alaska"?
4. In how many ways can the letters of the word "parallel" be arranged?
5. How does the number of arrangements change if a letter is added to "Alaska"?

**Tip**: When solving problems with repeated elements, always account for the repetitions using factorials to avoid overcounting.

In how many ways can the letters in the word 'Alaska' be arranged?

Learn how to calculate the number of ways to arrange the letters in the word 'Alaska'. This problem involves permutations with repeated elements, using the factorial formula. Suitable for students in Grades 8-10.

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