Key Concepts and Formulas

• Permutations with Repetitions: The number of distinct permutations of $n$ objects, where $n_1$ are alike of one kind, $n_2$ are alike of another kind, ..., and $n_k$ are alike of the $k^{th}$ kind, is given by: $\frac{n!}{n_1! n_2! \dots n_k!}$
• Complementary Principle: The number of ways to satisfy a condition is the total number of ways minus the number of ways the condition is not satisfied. $\text{Number of desired arrangements} = \text{Total number of arrangements} - \text{Number of arrangements where the condition IS met}$

Step 1: Calculate the Total Number of Distinct Words that can be Formed

We need to find the total number of arrangements of the letters in the word "LETTER" without any restrictions. This will be used as the total number of arrangements in the Complementary Principle.

The word "LETTER" has 6 letters: L, E, T, T, E, R. The letter 'T' appears twice and the letter 'E' appears twice.

Using the formula for permutations with repetitions, where $n = 6$, $n_T = 2$, and $n_E = 2$:

$\text{Total arrangements} = \frac{6!}{2! \cdot 2!}$ $= \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1) \times (2 \times 1)}$ $= \frac{720}{4}$ $= 180$

Therefore, there are 180 total distinct arrangements of the letters in "LETTER".

Step 2: Calculate the Number of Words where the Vowels (E, E) Always Come Together

We want to find the number of arrangements where the two 'E's are always adjacent. This will be subtracted from the total number of arrangements to find the arrangements where the vowels are not together.

Treat the two 'E's as a single block, "EE". Now we have 5 units to arrange: L, T, T, R, and (EE). Among these, 'T' is repeated twice.

Using the formula for permutations with repetitions, where $n = 5$ and $n_T = 2$:

$\text{Arrangements with vowels together} = \frac{5!}{2!}$ $= \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1}$ $= \frac{120}{2}$ $= 60$

Therefore, there are 60 arrangements where the two 'E's are together.

Step 3: Calculate the Number of Words where the Vowels Never Come Together

Apply the Complementary Principle to find the number of arrangements where the vowels are not together.

$\text{Words with vowels not together} = \text{Total arrangements} - \text{Arrangements with vowels together}$ $= 180 - 60$ $= 120$

Therefore, the number of words that can be formed from the letters of "LETTER" in which the vowels never come together is 120.

Common Mistakes and Tips to Avoid

• Forgetting Repetitions: Always account for repeated letters by dividing by the factorial of their counts.
• Block Arrangement: When grouping letters into a block, consider if the letters within the block are identical or distinct. If identical (like "EE"), there is no internal arrangement to consider. If distinct (like "AE"), you must multiply by the factorial of the block size.
• Complementary Principle Setup: Make sure you correctly identify the "total arrangements" and the arrangements where the condition is met.

Summary

We found the number of words where the vowels in "LETTER" never come together by first calculating the total number of arrangements of the letters in "LETTER", then calculating the number of arrangements where the vowels are together, and finally subtracting the latter from the former using the Complementary Principle. This gave us the final answer of 120.

The final answer is \boxed{120}. Since the options provided do not contain the correct answer, there might be an error in the options or the correct answer provided. The correct answer derived is 120.