How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent? (A) `7.^6C_4.^8C_4` (B) `8.^6C_4.^7C_4` (C) `6.7.^8C_4` (D) `6.8.^7C_4`

Options

A. `6*7*.^(8)C_(4)`
B. `6*8*.^(7)C_(4)`
C. `7*.^(6)C_(4)*.^(8)C_(4)`
D. `8*.^(6)C_(4)*.^(7)C_(4)`

Explanation

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

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

The total number of letters is 11.

### Step 2: Arrange the letters excluding 'S'
First, we will arrange the letters without the 'S's. The letters we have are M, I, I, I, I, P, P.

The total number of ways to arrange these 7 letters is given by the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{7!}{4! \cdot 2!} \]
Where:
- \(7!\) is the factorial of the total letters (M, I, I, I, I, P, P).
- \(4!\) accounts for the indistinguishable I's.
- \(2!\) accounts for the indistinguishable P's.

Calculating this:
\[ \frac{7!}{4! \cdot 2!} = \frac{5040}{24 \cdot 2} = \frac{5040}{48} = 105 \]

### Step 3: Determine positions for 'S's
Now we need to place the 4 'S's such that no two 'S's are adjacent.

After arranging the 7 letters (M, I, I, I, I, P, P), we create gaps where we can place 'S's. The arrangement of the 7 letters creates 8 potential gaps (before the first letter, between letters, and after the last letter).

For example, if we arrange the letters as:
_ M _ I _ I _ I _ I _ P _ P _
This creates 8 gaps (indicated by underscores).

### Step 4: Choose gaps for 'S's
We need to choose 4 out of these 8 gaps to place the 'S's. The number of ways to choose 4 gaps from 8 is given by the combination formula:
\[ \binom{8}{4} \]

Calculating this:
\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} = \frac{40320}{24 \cdot 24} = \frac{40320}{576} = 70 \]

### Step 5: Calculate the total arrangements
Now, we multiply the number of arrangements of the letters without 'S's by the number of ways to place the 'S's:
\[ \text{Total arrangements} = 105 \cdot 70 = 7350 \]

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