How many different words can be formed by jumbling the letters in the words MISSISSIPPI in which no two S are adjacent?

Options

A. ` ""^(6)C_ 4 . ""^(7)C_ 4 `
B. ` 6.8 ""^(8)C_ 4 `
C. `6.8 . ""^(7)C_4`
D. ` 7. ""^(6) C_4.""^(8)C_4`

Answer

To solve the problem of finding 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: Identify the letters and their frequencies
The word "MISSISSIPPI" consists of the following letters:
- M: 1
- I: 4
- S: 4
- P: 2

Total letters = 11.

### Step 2: Calculate arrangements without S
First, we will calculate the arrangements of the letters without considering S. The letters we will use are M, I, I, I, I, P, P (which totals to 7 letters).

The number of arrangements of these letters is given by the formula for permutations of multiset:
\[ \text{Arrangements without S} = \frac{7!}{4! \cdot 2!} \]
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.

### Step 3: Calculate the arrangements
Calculating the arrangements:
\[ 7! = 5040 \]
\[ 4! = 24 \]
\[ 2! = 2 \]
Thus,
\[ \text{Arrangements without S} = \frac{5040}{24 \cdot 2} = \frac{5040}{48} = 105. \]

### Step 4: Determine the positions for S
Next, we need to determine where we can place the S's. After arranging the letters M, I, I, I, I, P, P, we have the following arrangement:
- M, I, I, I, I, P, P

This arrangement creates 8 potential slots for placing S's:
- _ M _ I _ I _ I _ I _ P _ P _

### Step 5: Choose positions for S
We need to select 4 out of these 8 slots to place the S's. The number of ways to choose 4 slots from 8 is given by the combination formula:
\[ \text{Ways to choose positions for S} = \binom{8}{4}. \]

### Step 6: Calculate the combination
Calculating the combination:
\[ \binom{8}{4} = \frac{8!}{4! \cdot (8-4)!} = \frac{8!}{4! \cdot 4!} = \frac{40320}{24 \cdot 24} = 70. \]

### Step 7: Calculate the total arrangements
Finally, we multiply the number of arrangements without S by the number of ways to choose positions for S:
\[ \text{Total arrangements} = 105 \cdot 70 = 7350. \]

### Final Answer
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**.