How many different words can be formed by jumbling the letters of the word 'MISSISSIPPI' in which no two S are together ?

Options

A. `7.""^(6)C_(4).""^(8)C_(4)`
B. `8.""C_(4).""^(7)C_(4)`
C. `6.7.""^(8)C_(4)`
D. `6.8.""^(7)C_(4)`

Answer

To solve the problem of how many different words can be formed by jumbling the letters of the word "MISSISSIPPI" such that no two S's are together, we can follow these steps:

### Step 1: Count the letters in "MISSISSIPPI"
The word "MISSISSIPPI" consists of the following letters:
- M: 1
- I: 4
- S: 4
- P: 2

### Step 2: Calculate the total number of letters
The total number of letters in "MISSISSIPPI" is 11.

### Step 3: Arrange the letters excluding S
First, we will arrange the letters excluding S. The letters we have are M, I, I, I, I, P, P.

The number of arrangements of these letters can be calculated using the formula for permutations of multiset:

\[ \text{Number of arrangements} = \frac{n!}{n_1! \cdot n_2! \cdot n_3!} \]

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

In this case:

\[ \text{Number of arrangements} = \frac{7!}{1! \cdot 4! \cdot 2!} \]

Calculating this gives:

\[ = \frac{5040}{1 \cdot 24 \cdot 2} = \frac{5040}{48} = 105 \]

### Step 4: Determine the positions for S
When we arrange the letters M, I, I, I, I, P, P, we create gaps where we can place the S's. The arrangement creates 8 possible gaps (before and after each letter, including the ends).

For example, if we arrange the letters as M I I I I P P, the gaps are represented as follows:

_ M _ I _ I _ I _ I _ P _ P _

### Step 5: Choose positions for S
We need to choose 4 gaps from these 8 to place the S's. The number of ways to choose 4 gaps from 8 is given by the combination formula:

\[ \text{Number of ways to choose gaps} = \binom{8}{4} \]

Calculating this gives:

\[ \binom{8}{4} = \frac{8!}{4! \cdot (8-4)!} = \frac{8 \cdot 7 \cdot 6 \cdot 5}{4 \cdot 3 \cdot 2 \cdot 1} = 70 \]

### Step 6: Total arrangements
Finally, we multiply the number of arrangements of the letters excluding S by the number of ways to choose the gaps for S:

\[ \text{Total arrangements} = \text{Arrangements of non-S letters} \times \text{Ways to choose gaps} \]

\[ = 105 \times 70 = 7350 \]

### Final Answer
Thus, the total number of different words that can be formed by jumbling the letters of "MISSISSIPPI" such that no two S's are together is **7350**.

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