The number of arrangements that can be made out of the letters of the word SUCCESS so that all S do not come together is

Options

A. 60
B. 120
C. 360
D. 420

Answer

To solve the problem of finding the number of arrangements of the letters in the word "SUCCESS" such that not all 'S' letters come together, we can follow these steps:

### Step 1: Calculate the Total Arrangements
First, we need to find the total number of arrangements of the letters in the word "SUCCESS".

The word "SUCCESS" has 7 letters in total:
- S appears 3 times,
- C appears 2 times,
- U appears 1 time,
- E appears 1 time.

The formula for the total 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.

Thus, we have:

\[ \text{Total arrangements} = \frac{7!}{3! \times 2! \times 1! \times 1!} \]

Calculating this:

\[ 7! = 5040 \]
\[ 3! = 6, \quad 2! = 2, \quad 1! = 1 \]

Now substituting these values into the formula:

\[ \text{Total arrangements} = \frac{5040}{6 \times 2 \times 1 \times 1} = \frac{5040}{12} = 420 \]

### Step 2: Calculate the Arrangements with All 'S' Together
Next, we need to calculate the arrangements where all 'S' letters come together. We can treat the three 'S' letters as a single unit/block.

So, we can represent the letters as follows:
- Block of 'SSS', C, C, U, E

This gives us a total of 5 units to arrange:
- SSS, C, C, U, E

Now, we can calculate the arrangements of these 5 units:

\[ \text{Arrangements with S together} = \frac{5!}{2!} \]

Calculating this:

\[ 5! = 120, \quad 2! = 2 \]

Thus,

\[ \text{Arrangements with S together} = \frac{120}{2} = 60 \]

### Step 3: Calculate the Arrangements with 'S' Not Together
Finally, to find the arrangements where not all 'S' letters are together, we subtract the arrangements where all 'S' are together from the total arrangements:

\[ \text{Arrangements with S not together} = \text{Total arrangements} - \text{Arrangements with S together} \]

Substituting the values we calculated:

\[ \text{Arrangements with S not together} = 420 - 60 = 360 \]

### Final Answer
The number of arrangements of the letters of the word "SUCCESS" such that not all 'S' letters come together is **360**.

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