To find the number of ways the letters of the word "ACCLAIM" can be rearranged such that the vowels always appear together, we can follow these steps:
### Step 1: Identify the vowels and consonants
The word "ACCLAIM" consists of the letters A, C, C, L, A, I, M. The vowels in this word are A, A, and I.
### Step 2: Treat the vowels as a single unit
Since we want the vowels to appear together, we can treat the group of vowels (AAI) as a single unit. Thus, we can represent the word as:
- (AAI), C, C, L, M
This gives us a total of 5 units to arrange: (AAI), C, C, L, M.
### Step 3: Calculate the arrangements of the units
Now, we need to find the number of ways to arrange these 5 units. Since the letter C is repeated, we will use the formula for permutations of multiset:
\[
\text{Number of arrangements} = \frac{n!}{p_1! \cdot p_2! \cdots}
\]
where \( n \) is the total number of items to arrange, and \( p_1, p_2, \ldots \) are the frequencies of the repeated items.
Here, we have:
- Total units (n) = 5 (units: (AAI), C, C, L, M)
- The letter C is repeated 2 times.
Thus, the number of arrangements is:
\[
\text{Arrangements of units} = \frac{5!}{2!}
\]
### Step 4: Calculate the arrangements of the vowels
Next, we need to arrange the vowels within the unit (AAI). The vowels A, A, and I can be arranged as:
\[
\text{Arrangements of vowels} = \frac{3!}{2!}
\]
where the 2! accounts for the repetition of A.
### Step 5: Combine the results
Now, we multiply the number of arrangements of the units by the number of arrangements of the vowels:
\[
\text{Total arrangements} = \left(\frac{5!}{2!}\right) \times \left(\frac{3!}{2!}\right)
\]
### Step 6: Compute the values
Calculating the factorials:
- \( 5! = 120 \)
- \( 3! = 6 \)
- \( 2! = 2 \)
Now substituting these values:
\[
\text{Total arrangements} = \left(\frac{120}{2}\right) \times \left(\frac{6}{2}\right) = 60 \times 3 = 180
\]
### Final Answer
Thus, the total number of ways to rearrange the letters of the word "ACCLAIM" such that the vowels always appear together is **180**.
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