What is the probability that four S's come consecutively in the word MISSISSIPPI?

Options

A. `4/165`
B. `4/135`
C. `24/165`
D. none of these

Answer

To find the probability that four S's come consecutively in the word "MISSISSIPPI", we will follow these steps:

### Step 1: Determine the total arrangements of the letters in "MISSISSIPPI".

The word "MISSISSIPPI" consists of 11 letters: M, I, S, S, I, S, S, I, P, P, I.

The frequency of each letter is:
- M: 1
- I: 4
- S: 4
- P: 2

The total arrangements can be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots} \]

Where \( n \) is the total number of letters, and \( n_1, n_2, n_3, \ldots \) are the frequencies of each distinct letter.

Thus,

\[ \text{Total arrangements} = \frac{11!}{1! \cdot 4! \cdot 4! \cdot 2!} \]

### Step 2: Calculate the total arrangements.

Calculating \( 11! \):

\[ 11! = 39916800 \]

Calculating the factorials of the frequencies:

\[ 1! = 1, \quad 4! = 24, \quad 2! = 2 \]

Now substituting these values into the formula:

\[ \text{Total arrangements} = \frac{39916800}{1 \cdot 24 \cdot 24 \cdot 2} = \frac{39916800}{1152} = 34650 \]

### Step 3: Determine the favorable arrangements where all S's are together.

If we treat the four S's as a single unit, we can denote it as "S4". The letters we now have are:

- S4, M, I, I, I, P, P (total of 7 units)

Now we calculate the arrangements of these 7 units:

The frequency of the letters now is:
- S4: 1
- M: 1
- I: 3
- P: 2

Thus, the arrangements can be calculated as:

\[ \text{Favorable arrangements} = \frac{7!}{1! \cdot 3! \cdot 2!} \]

### Step 4: Calculate the favorable arrangements.

Calculating \( 7! \):

\[ 7! = 5040 \]

Calculating the factorials of the frequencies:

\[ 3! = 6, \quad 2! = 2 \]

Now substituting these values into the formula:

\[ \text{Favorable arrangements} = \frac{5040}{1 \cdot 6 \cdot 2} = \frac{5040}{12} = 420 \]

### Step 5: Calculate the probability.

The probability \( P \) that four S's come consecutively is given by the ratio of favorable outcomes to total outcomes:

\[ P = \frac{\text{Favorable arrangements}}{\text{Total arrangements}} = \frac{420}{34650} \]

### Step 6: Simplify the probability.

To simplify \( \frac{420}{34650} \):

\[ P = \frac{420 \div 30}{34650 \div 30} = \frac{14}{1155} \]

Further simplifying:

\[ P = \frac{4}{165} \]

### Final Answer:

Thus, the probability that four S's come consecutively in the word "MISSISSIPPI" is:

\[ \boxed{\frac{4}{165}} \]