How many words can be formed using all the letters of the word 'NATION' so that all the three vowels should never come together ?

Options

A. 354
B. 348
C. 288
D. None of these

Explanation

To solve the problem of how many words can be formed using all the letters of the word "NATION" such that all three vowels (A, I, O) never come together, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in "NATION".
The word "NATION" consists of 6 letters where 'N' appears twice. The total arrangements can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

Where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

In "NATION":
- Total letters, \( n = 6 \)
- 'N' appears 2 times.

Thus, the total arrangements are:

\[ \text{Total arrangements} = \frac{6!}{2!} = \frac{720}{2} = 360 \]

### Step 2: Calculate the arrangements where all vowels come together.
To find the arrangements where the vowels A, I, O come together, we can treat the three vowels as a single unit or block. Therefore, we have the following blocks to arrange: (AIO), N, T, N.

This gives us a total of 4 blocks (AIO, N, T, N). The arrangements of these blocks can be calculated as:

\[ \text{Arrangements of blocks} = \frac{4!}{2!} = \frac{24}{2} = 12 \]

Now, within the block (AIO), the vowels can be arranged among themselves in:

\[ 3! = 6 \text{ ways} \]

Thus, the total arrangements where all vowels come together is:

\[ \text{Total arrangements with vowels together} = 12 \times 6 = 72 \]

### Step 3: Calculate the arrangements where the vowels do not come together.
To find the arrangements where the vowels do not come together, we subtract the arrangements where they do come together from the total arrangements:

\[ \text{Arrangements where vowels do not come together} = \text{Total arrangements} - \text{Arrangements with vowels together} \]

Substituting the values we calculated:

\[ \text{Arrangements where vowels do not come together} = 360 - 72 = 288 \]

### Final Answer:
The number of words that can be formed using all the letters of the word "NATION" such that all three vowels never come together is **288**.

---