In how many different ways can the letters of the word 'SALOON' be arranged If the two O's must not come together?

Answer

To solve the problem of arranging the letters of the word 'SALOON' such that the two O's do not come together, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in 'SALOON'.
The word 'SALOON' consists of 6 letters where the letter 'O' is repeated twice. The formula for the arrangements of letters when there are repetitions is given by:

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

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

In this case:
- Total letters, \( n = 6 \) (S, A, L, O, O, N)
- The letter O is repeated 2 times.

Thus, the total arrangements can be calculated as:

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

### Step 2: Calculate the arrangements where the two O's are together.
To find the arrangements where the two O's are together, we can treat the two O's as a single unit or block. This gives us the following letters to arrange: (OO), S, A, L, N, which totals to 5 units.

Now, we calculate the arrangements of these 5 units:

\[ \text{Arrangements with OO together} = 5! = 120 \]

### Step 3: Calculate the arrangements where the two O's do not come together.
To find the arrangements where the two O's do not come together, we subtract the arrangements where the O's are together from the total arrangements.

\[ \text{Arrangements with O's not together} = \text{Total arrangements} - \text{Arrangements with OO together} \]

Substituting the values we calculated:

\[ \text{Arrangements with O's not together} = 360 - 120 = 240 \]

### Final Answer:
The number of different ways the letters of the word 'SALOON' can be arranged such that the two O's do not come together is **240**.

---