How many different ways can the letters of "millennium" be​ arranged?

To find how many different ways the letters of "millennium" can be arranged, we use the concept of permutations of multiset letters.

### Step 1: Identify the total number of letters.
"millennium" consists of 10 letters.

### Step 2: Identify the repeated letters.
- "m" appears 2 times.
- "i" appears 2 times.
- "l" appears 2 times.
- "n" appears 2 times.
- "e" and "u" appear once each.

### Step 3: Use the permutation formula for multiset letters.
The formula for the number of ways to arrange letters with repetitions is:

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

where:
- \(n\) is the total number of letters.
- \(k_1, k_2, \dots, k_r\) are the frequencies of the repeated letters.

For "millennium":
\[
n = 10, \quad k_1 = 2 \, (m), \quad k_2 = 2 \, (i), \quad k_3 = 2 \, (l), \quad k_4 = 2 \, (n)
\]

Thus, the formula becomes:
\[
\text{Number of permutations} = \frac{10!}{2! \times 2! \times 2! \times 2!}
\]

### Step 4: Calculate the factorial values.
- \(10! = 3,628,800\)
- \(2! = 2\)

Now plug these into the formula:

\[
\text{Number of permutations} = \frac{3,628,800}{2 \times 2 \times 2 \times 2} = \frac{3,628,800}{16} = 226,800
\]

### Final Answer:
The letters of "millennium" can be arranged in **226,800 different ways**.

Would you like more details or have any questions?

Here are some related questions you might explore:
1. How would the number of arrangements change if all letters were distinct?
2. What if the word "millennium" had three "l"s instead of two?
3. How many ways can you arrange only the consonants in "millennium"?
4. How many arrangements are there if we restrict the positions of vowels and consonants?
5. How many ways can you select 5 letters from "millennium"?

**Tip:** When calculating permutations with repeated elements, always account for how often each letter repeats!

How many different ways can the letters of 'millennium' be arranged?

Learn how to calculate the number of ways the letters of 'millennium' can be arranged, using permutations of multiset letters and factorials. This detailed explanation includes identifying repeated letters and applying the permutation formula for multisets. Suitable for students in Grades 9-12.

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