In how many ways can the letters of the word ASSASSINATION be arranged so that all the S's occur together ?

Answer

To find the number of ways to arrange the letters of the word "ASSASSINATION" such that all the S's occur together, we can follow these steps:

### Step 1: Count the total letters in the word.
The word "ASSASSINATION" has a total of 13 letters.

### Step 2: Treat all S's as a single unit.
Since we want all S's to occur together, we can treat the four S's as one single unit or letter. This means we will consider "SSSS" as one letter.

### Step 3: Count the remaining letters.
After treating the four S's as one unit, we have the following letters remaining:
- A
- A
- I
- N
- A
- T
- I
- O
- N

So, the letters we have now are: S (as one unit), A, A, I, N, A, T, I, N. This gives us a total of 10 units (1 S + 3 A's + 2 I's + 2 N's + 1 T).

### Step 4: Calculate the arrangements of these units.
The formula for the arrangements of n items where there are repetitions is given by:

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

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

In our case:
- Total units (n) = 10
- A appears 3 times
- I appears 2 times
- N appears 2 times

So, the number of arrangements is:

\[ \text{Number of arrangements} = \frac{10!}{3! \times 2! \times 2!} \]

### Step 5: Calculate the factorials.
Now we calculate the factorials:
- \( 10! = 3628800 \)
- \( 3! = 6 \)
- \( 2! = 2 \)

### Step 6: Substitute the values into the formula.
Substituting the values into the formula gives us:

\[ \text{Number of arrangements} = \frac{3628800}{6 \times 2 \times 2} = \frac{3628800}{24} = 151200 \]

### Final Answer:
Thus, the total number of ways to arrange the letters of the word "ASSASSINATION" such that all S's occur together is **151200**.

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