Permutations of JUXTAPOSED with Vowels Together

The question asks for the number of distinct permutations of the letters in the word JUXTAPOSED where the four vowels always stay together.

First, let's list the letters in the word JUXTAPOSED:

• J, U, X, T, A, P, O, S, E, D

There are a total of 10 letters in the word JUXTAPOSED.

Next, let's identify the vowels and consonants in JUXTAPOSED:

• Vowels: U, A, O, E (There are 4 vowels)
• Consonants: J, X, T, P, S, D (There are 6 consonants)

All the letters in JUXTAPOSED are distinct.

Calculating Permutations with Vowels as a Single Unit

The condition is that the four vowels must come together. To handle this, we can treat the block of four vowels (UAE) as a single unit or a single "letter".

Now, we are arranging the 6 consonants and this one block of vowels. So, we are arranging a total of $6 + 1 = 7$ objects.

The number of ways to arrange these 7 distinct objects is given by the factorial of 7, which is $7!$.

Number of arrangements of the 7 units (6 consonants + 1 vowel block) = $7!$

\(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)

Arrangements Within the Vowel Block

The four vowels (U, A, O, E) within their block can be arranged among themselves in any order. Since there are 4 distinct vowels, the number of ways to arrange them within their block is $4!$.

Number of arrangements of the 4 distinct vowels within the block = $4!$

\(4! = 4 \times 3 \times 2 \times 1 = 24\)

Total Distinct Permutations with Vowels Together

To find the total number of distinct permutations of JUXTAPOSED where the four vowels come together, we multiply the number of ways to arrange the 7 units by the number of ways to arrange the vowels within their block.

Total permutations = (Arrangement of 7 units) $\times$ (Arrangement of vowels within the block)

Total permutations = \(7! \times 4!\)

Total permutations = \(5040 \times 24\)

Let's perform the multiplication:

\(5040 \times 24 = 120960\)

Therefore, there are 120,960 distinct permutations of the letters in JUXTAPOSED where the four vowels come together.