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

Options

A. 10!
B. 14!/(4!)
C. 151200
D. 3628800

Answer

To solve the problem of arranging the letters of the word 'ASSASSINATION' such that all the 'S's are together, we can follow these steps:

### Step 1: Treat the 'S's as a single unit
Since we want all the 'S's to be together, we can treat them as one single unit or block. So instead of having 4 'S's, we can represent them as a single block, which we can denote as [SSSS].

### Step 2: Count the remaining letters
Now, we need to count the letters in the word 'ASSASSINATION' excluding the 'S's. The letters we have are:
- A: 3
- I: 2
- N: 2
- T: 1
- O: 1
- [SSSS]: 1 (the block of S's)

So, the letters we are arranging now are:
- A, A, A, I, I, N, N, T, O, [SSSS]

This gives us a total of 10 units to arrange (the 3 A's, 2 I's, 2 N's, 1 T, 1 O, and the block of S's).

### Step 3: Calculate the arrangements
The total number of arrangements of these 10 units, considering the repetitions, is given by the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \times \ldots} \]

Where:
- \( n \) is the total number of items to arrange,
- \( n_1, n_2, n_3, \ldots \) are the counts of each indistinguishable item.

In our case:
- Total units \( n = 10 \)
- A's = 3, I's = 2, N's = 2, T's = 1, O's = 1, [SSSS] = 1

So, we have:

\[ \text{Total arrangements} = \frac{10!}{3! \times 2! \times 2! \times 1! \times 1!} \]

### Step 4: Calculate the factorials
Now we calculate each factorial:

- \( 10! = 3628800 \)
- \( 3! = 6 \)
- \( 2! = 2 \)
- \( 1! = 1 \)

### Step 5: Substitute the values into the formula
Now substituting these values into our formula:

\[ \text{Total arrangements} = \frac{3628800}{6 \times 2 \times 2 \times 1 \times 1} \]
\[ = \frac{3628800}{24} \]
\[ = 151200 \]

### Final Answer
Thus, the total number of ways the letters of the word 'ASSASSINATION' can be arranged such that all the 'S's are together is **151200**.

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