There are 5 different Jeffrey Archer books, 3 different Sidney Sheldon books and 6 different John Grisham books. The number of ways in which at least one book can be given away is

Options

A. `2^(10) -1`
B. `2^(11)-1`
C. `2^(12)-1`
D. `2^(14)-1`

Answer

To find the number of ways in which at least one book can be given away from a collection of books by Jeffrey Archer, Sidney Sheldon, and John Grisham, we can follow these steps:

### Step-by-Step Solution:

1. **Count the Total Number of Books**:
- Jeffrey Archer books: 5
- Sidney Sheldon books: 3
- John Grisham books: 6
- Total number of books = 5 + 3 + 6 = 14

2. **Determine the Options for Each Book**:
- For each book, there are 2 options: either to give it away or to keep it.

3. **Calculate the Total Combinations**:
- Since there are 14 books, and each book has 2 options, the total number of combinations of giving away or keeping the books is \(2^{14}\).

4. **Exclude the Case Where No Books Are Given Away**:
- The total combinations calculated above includes the scenario where no books are given away (i.e., all books are kept). We need to subtract this case.
- There is only 1 way to keep all books (not giving away any), so we subtract 1 from the total combinations.

5. **Final Calculation**:
- The number of ways to give away at least one book = Total combinations - 1
- Therefore, the number of ways = \(2^{14} - 1\).

6. **Calculate \(2^{14}\)**:
- \(2^{14} = 16384\)
- So, \(16384 - 1 = 16383\).

### Final Answer:
The number of ways in which at least one book can be given away is **16383**.