The number of words that can be formed using all the letters of the word ⁠\(\text{EQUATION}\) so that all the vowels are together

The word "EQUATION" consists of 8 letters, with 5 vowels (E, U, A, I, O) and 3 consonants (Q, T, N).

### Step 1: Treating all vowels as a single unit
Since the problem asks that all vowels must be together, we can consider the 5 vowels (E, U, A, I, O) as one "block." Thus, we now have to arrange 4 units: the vowel block (V), and the consonants Q, T, and N.

### Step 2: Arranging the units
The total number of ways to arrange these 4 units (V, Q, T, N) is simply the number of permutations of these 4 units, which is:
\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]

### Step 3: Arranging the vowels within the block
Now, the vowels within the "vowel block" can be arranged among themselves. Since there are 5 vowels, the number of ways to arrange them is:
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Step 4: Total number of arrangements
The total number of words that can be formed where all the vowels are together is the product of the number of ways to arrange the 4 units and the number of ways to arrange the vowels within the block:
\[
4! \times 5! = 24 \times 120 = 2880
\]

Thus, the total number of words that can be formed is **2880**.

Would you like further explanation or have any other questions?

Here are 5 related questions for practice:
1. How many distinct words can be formed using all the letters of the word "EQUATION" without any restrictions?
2. If two specific vowels in "EQUATION" must always be together, how many distinct arrangements are possible?
3. What if only three out of the five vowels in "EQUATION" must be together? How many arrangements are there?
4. How many words can be formed where no vowels are adjacent in "EQUATION"?
5. If the word "EQUATION" is arranged such that vowels and consonants alternate, how many distinct words can be formed?

**Tip:** For problems involving constraints on groupings (like keeping vowels together), consider treating grouped items as a single unit first!

The number of words that can be formed using all the letters of the word 'EQUATION' so that all the vowels are together.

Find how many distinct words can be formed from the letters of 'EQUATION' with all vowels together. This solution involves combinatorics and factorial calculations, using concepts like permutations of grouped items. Suitable for high school students.

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