The total number of ways in which letters of the word ACCOST can be arranged so that the two C's never come together will be

Options

A. 120
B. 360
C. 240
D. 6 ! – 2 !

Answer

To find the total number of ways in which the letters of the word "ACCOST" can be arranged such that the two C's never come together, we can follow these steps:

### Step 1: Calculate the total arrangements of the letters in "ACCOST".
The word "ACCOST" consists of 6 letters where 'C' appears twice. The formula for the arrangement 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,
- \( p_1, p_2, \ldots \) are the frequencies of the repeated letters.

For "ACCOST":
- Total letters \( n = 6 \) (A, C, C, O, S, T)
- Frequency of C = 2

Thus, the total arrangements are:

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

### Step 2: Calculate the arrangements where the two C's are together.
To find the arrangements where the two C's are together, we can treat the two C's as a single unit or block. Therefore, we can think of the arrangement as:

- Block (CC), A, O, S, T

This gives us 5 units to arrange: (CC), A, O, S, T.

The number of arrangements of these 5 units is:

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

### Step 3: Calculate the arrangements where the two C's are never together.
To find the arrangements where the two C's are never together, we subtract the arrangements where the C's are together from the total arrangements:

\[ \text{Arrangements with C's never together} = \text{Total arrangements} - \text{Arrangements with CC together} \]

Substituting the values we calculated:

\[ \text{Arrangements with C's never together} = 360 - 120 = 240 \]

### Final Answer:
The total number of ways in which the letters of the word "ACCOST" can be arranged such that the two C's never come together is **240**.

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