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Step-by-Step Solution
Step 1: Identify the letters and their counts
The word “LETTER” consists of 6 letters in total: L, E, T, T, E, R. Among these:
Vowels: E, E (2 E’s)
Consonants: L, T, T, R
Step 2: Calculate the total number of arrangements of the letters
Since the word has 6 letters in total, but with repetitions (2 E’s and 2 T’s), the total number of distinct arrangements is given by:
\frac{6!}{2!\,2!} = \frac{720}{4} = 180
Step 3: Calculate the number of arrangements where the vowels (E, E) are together
Treat the pair of vowels (EE) as a single entity. Then we effectively have these entities to arrange: [EE], L, T, T, R.
That gives us 5 entities to arrange, where T still occurs twice. Thus, the number of such arrangements is:
\frac{5!}{2!} = \frac{120}{2} = 60
Step 4: Subtract the arrangements with vowels together from the total
We want the number of arrangements where the two vowels (E, E) never come together. Hence we subtract the number of arrangements with vowels together from the total arrangements:
Number of arrangements where vowels never come together
= 180 − 60 = 120
Final Answer: 120
