In how many arrangements of the word 'GOLDEN' will the vowels never occur together ?

Explanation

To solve the problem of how many arrangements of the word "GOLDEN" will have the vowels never occurring together, we can follow these steps:

### Step 1: Count the total arrangements of the word "GOLDEN".
The word "GOLDEN" consists of 6 distinct letters: G, O, L, D, E, N.

The total number of arrangements of these letters is given by the factorial of the number of letters:
\[ \text{Total arrangements} = 6! = 720 \]

### Step 2: Consider the vowels together.
The vowels in "GOLDEN" are O and E. To find the arrangements where the vowels are together, we can treat the vowels as a single unit or block.

So, we can represent the arrangement as:
- Block of vowels (OE)
- Consonants (G, L, D, N)

This gives us 5 units to arrange: (OE), G, L, D, N.

### Step 3: Calculate arrangements with vowels together.
The number of arrangements of these 5 units is:
\[ 5! = 120 \]
However, the block of vowels (OE) can also be arranged among themselves. Since there are 2 vowels, they can be arranged in:
\[ 2! = 2 \]

Thus, the total arrangements with the vowels together is:
\[ \text{Arrangements with vowels together} = 5! \times 2! = 120 \times 2 = 240 \]

### Step 4: Calculate arrangements where vowels do not occur together.
To find the arrangements where the vowels do not occur together, we subtract the arrangements where they are together from the total arrangements:
\[ \text{Arrangements where vowels do not occur together} = \text{Total arrangements} - \text{Arrangements with vowels together} \]
\[ = 720 - 240 = 480 \]

### Conclusion:
The number of arrangements of the word "GOLDEN" where the vowels O and E do not occur together is **480**.

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