The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
Answer: 36000
Step-by-step solution
Step 1: Identify total letters and vowels/consonants
First, we identify the total number of letters in the word "DAUGHTER". Then, we separate the letters into vowels and consonants. This helps us to categorize and count them for permutation calculations.
Step 2: Calculate total permutations
The total number of distinct words that can be formed using all the letters of "DAUGHTER" is the factorial of the total number of letters, since all letters are distinct. This gives us the total possible arrangements without any restrictions.
Step 3: Calculate permutations where all vowels come together
To find the number of words where all vowels come together, we treat the group of vowels (A, U, E) as a single block. Now we have 6 'items' to arrange (5 consonants + 1 vowel block). Additionally, the vowels within their block can be arranged in ways.
Step 4: Calculate permutations where vowels never come together
The number of words where all vowels never come together is found by subtracting the number of words where all vowels *do* come together from the total number of possible words. This is a common technique for 'never together' problems in permutations.
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