In how many ways can the letters of the word 'HEXAGON' be permuted? In how many words will the vowels be together?

Explanation

To solve the problem, we will break it down into two parts:

1. Finding the total number of permutations of the letters in the word "HEXAGON".
2. Finding the number of permutations where the vowels are together.

### Part 1: Total Permutations of the Word "HEXAGON"

1. **Identify the number of letters**:
The word "HEXAGON" consists of 7 letters.

2. **Calculate the total permutations**:
Since all the letters are distinct, the total number of permutations can be calculated using the factorial of the number of letters.
\[ \text{Total permutations} = 7! = 5040 \]

### Part 2: Permutations with Vowels Together

1. **Identify the vowels**:
The vowels in "HEXAGON" are E, A, and O. There are 3 vowels.

2. **Consider the vowels as a single unit**:
If we treat the group of vowels (E, A, O) as one single letter, we can represent it as (EAO). Now, we have the following units to arrange:
- (EAO), H, X, G, N
This gives us a total of 5 units to arrange.

3. **Calculate the permutations of these units**:
The number of ways to arrange these 5 units is given by:
\[ 5! = 120 \]

4. **Calculate the arrangements of the vowels**:
The vowels (E, A, O) can be arranged among themselves in:
\[ 3! = 6 \text{ ways} \]

5. **Calculate the total arrangements with vowels together**:
To find the total number of arrangements where the vowels are together, we multiply the arrangements of the units by the arrangements of the vowels:
\[ \text{Total arrangements with vowels together} = 5! \times 3! = 120 \times 6 = 720 \]

### Final Answers:
- The total number of permutations of the letters in "HEXAGON" is **5040**.
- The number of permutations where the vowels are together is **720**.