To solve the problem of arranging the letters of the word "IMAGINARY" such that all the vowels never come together, we can follow these steps:
### Step 1: Identify the letters and vowels
The word "IMAGINARY" consists of 9 letters: I, M, A, G, I, N, A, R, Y. The vowels in the word are I, A, I, A.
### Step 2: Calculate the total arrangements of the letters
To find the total arrangements of the letters, we use the formula for permutations of multiset:
\[
\text{Total arrangements} = \frac{n!}{p_1! \times p_2!}
\]
Where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots \) are the frequencies of the repeated letters.
Here, we have:
- Total letters \( n = 9 \)
- I appears 2 times,
- A appears 2 times.
Thus, the total arrangements are:
\[
\text{Total arrangements} = \frac{9!}{2! \times 2!}
\]
### Step 3: Calculate \( 9! \) and \( 2! \)
Calculating the factorials:
- \( 9! = 362880 \)
- \( 2! = 2 \)
Now substituting the values:
\[
\text{Total arrangements} = \frac{362880}{2 \times 2} = \frac{362880}{4} = 90720
\]
### Step 4: Calculate arrangements where vowels are together
Next, we treat the vowels (I, A, I, A) as a single unit. This gives us the units: (IAIA), M, G, N, R, Y. This results in 6 units.
### Step 5: Calculate arrangements of these units
The arrangements of these 6 units, where the vowels are treated as one unit, is:
\[
\text{Arrangements} = \frac{6!}{2! \times 2!}
\]
Where:
- \( 6! = 720 \)
- The vowels I and A repeat.
Calculating:
\[
\text{Arrangements} = \frac{720}{2 \times 2} = \frac{720}{4} = 180
\]
### Step 6: Calculate arrangements of vowels within the unit
The arrangements of the vowels (I, A, I, A) is:
\[
\text{Vowel arrangements} = \frac{4!}{2! \times 2!} = \frac{24}{4} = 6
\]
### Step 7: Total arrangements where vowels are together
Now, we multiply the arrangements of the units by the arrangements of the vowels:
\[
\text{Total arrangements (vowels together)} = 180 \times 6 = 1080
\]
### Step 8: Calculate arrangements where vowels are not together
Finally, to find the arrangements where the vowels are not together, we subtract the arrangements where they are together from the total arrangements:
\[
\text{Arrangements (vowels not together)} = \text{Total arrangements} - \text{Total arrangements (vowels together)}
\]
\[
= 90720 - 1080 = 89640
\]
### Final Answer
The number of different ways the letters of the word "IMAGINARY" can be arranged such that all the vowels never come together is **89640**.
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