Total number of ways in which the letters of the word 'MISSISSIPPI' can be arranged, so that any two S's are separated, is equal to

Options

A. 7350
B. 3650
C. 6250
D. 1261

Answer

To find the total number of ways to arrange the letters of the word "MISSISSIPPI" such that no two 'S's are adjacent, we can follow these steps:

### Step 1: Identify the letters in "MISSISSIPPI"
The letters in "MISSISSIPPI" are:
- M: 1
- I: 4
- S: 4
- P: 2

### Step 2: Arrange the letters excluding 'S'
First, we will arrange the letters excluding 'S'. The letters are M, I, I, I, I, P, P.

The total number of arrangements of these letters can be calculated using the formula for permutations of multiset:
\[ \text{Total arrangements} = \frac{n!}{n_1! \times n_2! \times n_3! \ldots} \]
where \(n\) is the total number of letters, and \(n_1, n_2, \ldots\) are the frequencies of each distinct letter.

Here, we have:
- Total letters = 7 (M, I, I, I, I, P, P)
- Frequencies: 1 M, 4 I's, 2 P's

Thus, the number of arrangements is:
\[ \text{Arrangements} = \frac{7!}{1! \times 4! \times 2!} = \frac{5040}{1 \times 24 \times 2} = \frac{5040}{48} = 105 \]

### Step 3: Determine the gaps for 'S'
When we arrange these 7 letters (M, I, I, I, I, P, P), we create gaps where we can place the 'S's. The arrangement of 7 letters creates 8 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, the gaps would be:
_ M _ I _ I _ I _ I _ P _ P _

### Step 4: Choose gaps for 'S'
We need to choose 4 gaps out of the 8 available gaps to place the 'S's. The number of ways to choose 4 gaps from 8 is given by the combination formula:
\[ \text{Ways to choose gaps} = \binom{8}{4} = \frac{8!}{4! \times (8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

### Step 5: Calculate the total arrangements
Now, we multiply the number of arrangements of the letters excluding 'S' by the number of ways to choose the gaps for 'S':
\[ \text{Total arrangements with separated S's} = \text{Arrangements of letters} \times \text{Ways to choose gaps} = 105 \times 70 = 7350 \]

### Final Answer
The total number of ways in which the letters of the word "MISSISSIPPI" can be arranged such that no two 'S's are adjacent is **7350**.

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