In how many ways can the letters of the word 'GLOOMY' be arranged so that the two O's should not be together?

Options

A. 240
B. 480
C. 60
D. 720

Answer

To solve the problem of arranging the letters of the word "GLOOMY" such that the two O's are not together, we can follow these steps:

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

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

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

For "GLOOMY":
- Total letters, \( n = 6 \)
- The letter O appears twice, so \( p_1 = 2 \)

Thus, the total arrangements are:

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

### 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 entity or "block". This reduces the problem to arranging the letters G, L, M, Y, and the block OO.

Now we have 5 entities to arrange: G, L, M, Y, OO.

The number of arrangements of these 5 entities is:

\[ 5! = 120 \]

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

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

Substituting the values we calculated:

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

### Final Answer
Thus, the number of ways to arrange the letters of the word "GLOOMY" such that the two O's are not together is **240**.

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