There are `10` different books in a shelf. The number of ways in which three books can be selected so that exactly two of them are consecutive is

Options

A. `60`
B. `54`
C. `56`
D. `36`

Explanation

To solve the problem of selecting three books from a shelf of 10 different books such that exactly two of them are consecutive, we can break down the solution into two cases:

### Step-by-Step Solution:

**Step 1: Identify the total number of books.**
- We have 10 different books on the shelf.

**Step 2: Define the condition for selection.**
- We need to select 3 books such that exactly 2 of them are consecutive.

**Step 3: Case 1 - Consecutive books at the terminal ends.**
- The consecutive books can be at the start or the end of the shelf.
- The pairs of consecutive books can be (Book 1, Book 2), (Book 2, Book 3), ..., (Book 9, Book 10).
- In this case, the pairs can be:
- (Book 1, Book 2)
- (Book 9, Book 10)

- This gives us 2 options for placing the consecutive books at the terminal ends.

**Step 4: Select the third book.**
- After selecting 2 consecutive books, we need to select the third book such that it is not consecutive to the selected pair.
- If we have selected (Book 1, Book 2), the third book can be chosen from Books 3 to 10, which gives us 8 options.
- If we have selected (Book 9, Book 10), the third book can be chosen from Books 1 to 8, which also gives us 8 options.

- Therefore, for Case 1, the total number of ways is:
\[ 2 \text{ (choices for pairs)} \times 8 \text{ (choices for third book)} = 16 \text{ ways} \]

**Step 5: Case 2 - Consecutive books not at the terminal ends.**
- The pairs of consecutive books can be:
- (Book 2, Book 3)
- (Book 3, Book 4)
- (Book 4, Book 5)
- (Book 5, Book 6)
- (Book 6, Book 7)
- (Book 7, Book 8)
- (Book 8, Book 9)

- This gives us 8 pairs of consecutive books.

**Step 6: Select the third book.**
- After selecting a pair, we need to choose the third book such that it is not consecutive to the selected pair.
- For each pair, the third book cannot be one of the two consecutive books and cannot be the book immediately adjacent to them. This leaves us with 6 options for the third book.

- Therefore, for Case 2, the total number of ways is:
\[ 8 \text{ (choices for pairs)} \times 6 \text{ (choices for third book)} = 48 \text{ ways} \]

**Step 7: Combine the results from both cases.**
- Total ways = Ways from Case 1 + Ways from Case 2
\[ 16 + 48 = 64 \text{ ways} \]

### Final Answer:
The total number of ways to select 3 books such that exactly 2 of them are consecutive is **64**.