Number of ways in which 15 different books can be arraged on a shelf so that two particular books shall not be together is

Options

A. `14xx 15!`
B. `13xx14!`
C. 14! `xx` 15!
D. `(15!)^(2)`

Answer

To solve the problem of arranging 15 different books on a shelf such that two particular books are not together, we can follow these steps:

### Step 1: Calculate the total arrangements of 15 books
The total number of ways to arrange 15 different books is given by the factorial of the number of books. Thus, the total arrangements can be calculated as:

\[ \text{Total arrangements} = 15! \]

### Step 2: Calculate the arrangements where the two particular books are together
To find the number of arrangements where the two particular books are together, we can treat these two books as a single unit or "block." This means we now have 14 units to arrange (the block of 2 books plus the other 13 books).

The number of ways to arrange these 14 units is:

\[ \text{Arrangements with two books together} = 14! \]

Since the two books can be arranged among themselves in 2 different ways (Book A can be first or Book B can be first), we multiply by 2:

\[ \text{Total arrangements with two books together} = 14! \times 2 \]

### Step 3: Subtract the arrangements where the two books are together from the total arrangements
Now, we subtract the number of arrangements where the two particular books are together from the total arrangements to find the arrangements where they are not together:

\[ \text{Arrangements where two books are not together} = 15! - (14! \times 2) \]

### Step 4: Simplify the expression
We can factor out \(14!\) from the expression:

\[ 15! = 15 \times 14! \]

Thus, we can rewrite the expression as:

\[ \text{Arrangements where two books are not together} = 15 \times 14! - 2 \times 14! \]

Factoring out \(14!\):

\[ = 14! \times (15 - 2) = 14! \times 13 \]

### Final Answer
Therefore, the number of ways in which 15 different books can be arranged on a shelf so that the two particular books are not together is:

\[ \text{Final Answer} = 13 \times 14! \]

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