To solve the problem of finding the total number of words that can be formed from the letters of the word "LETTER" such that no two vowels are together, we can follow these steps:
### Step 1: Identify the letters and their counts
The word "LETTER" consists of the letters: L, E, T, T, E, R.
- Vowels: E, E (2 vowels)
- Consonants: L, T, T, R (4 consonants)
### Step 2: Calculate the total arrangements of the letters without restrictions
The total number of arrangements of the letters in "LETTER" can be calculated using the formula for permutations of multiset:
\[
\text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}
\]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the counts of each repeated letter.
In "LETTER":
- Total letters = 6 (L, E, T, T, E, R)
- E appears 2 times, T appears 2 times.
Thus,
\[
\text{Total arrangements} = \frac{6!}{2! \times 2!} = \frac{720}{4} = 180
\]
### Step 3: Calculate arrangements where the two vowels are together
To find the arrangements where the two vowels (E, E) are together, we can treat them as a single unit or block. This gives us the following letters to arrange:
- Block (EE), L, T, T, R
Now we have 5 units to arrange: (EE), L, T, T, R.
The arrangements can be calculated as:
\[
\text{Arrangements with vowels together} = \frac{5!}{2!} = \frac{120}{2} = 60
\]
(The 2! accounts for the repeated T's.)
### Step 4: Calculate arrangements where no two vowels are together
To find the arrangements where no two vowels are together, we subtract the arrangements where the vowels are together from the total arrangements:
\[
\text{Arrangements with no two vowels together} = \text{Total arrangements} - \text{Arrangements with vowels together}
\]
\[
= 180 - 60 = 120
\]
### Final Answer
The total number of words that can be formed from the letters of the word "LETTER" such that no two vowels are together is **120**.
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