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 will break the solution into two cases:

**Case 1: The two consecutive books are at the ends of the shelf.**

1. Identify the positions of the two consecutive books. They can either be:
- Books at positions 1 and 2 (A1, A2)
- Books at positions 9 and 10 (A9, A10)

2. For each of these scenarios, we need to select the third book from the remaining books.
- If we select A1 and A2, the remaining books are A3, A4, A5, A6, A7, A8, which gives us 7 options for the third book.
- If we select A9 and A10, the remaining books are A1, A2, A3, A4, A5, A6, A7, which also gives us 7 options for the third book.

3. Therefore, for Case 1, the total number of ways to select the books is:
\[ 2 \text{ (for the two end cases)} \times 7 \text{ (options for the third book)} = 14 \]

**Case 2: The two consecutive books are not at the ends of the shelf.**

1. The two consecutive books can be selected from the following pairs:
- (A2, A3), (A3, A4), (A4, A5), (A5, A6), (A6, A7), (A7, A8), (A8, A9)

This gives us a total of 7 pairs.

2. For each pair selected, we need to choose the third book from the remaining books. The third book cannot be adjacent to the two consecutive books.
- For example, if we select A2 and A3, the remaining books are A1, A4, A5, A6, A7, A8, A9, A10 (8 options).
- If we select A3 and A4, the remaining books are A1, A2, A5, A6, A7, A8, A9, A10 (8 options).
- Continuing this way, we find that for each pair (A2, A3) to (A8, A9), we will have 6 options for the third book.

3. Therefore, for Case 2, the total number of ways to select the books is:
\[ 7 \text{ (pairs)} \times 6 \text{ (options for the third book)} = 42 \]

**Final Calculation:**
To find the total number of ways to select the three books such that exactly two are consecutive, we add the results from both cases:
\[ \text{Total} = \text{Case 1} + \text{Case 2} = 14 + 42 = 56 \]

Thus, the total number of ways to select three books such that exactly two of them are consecutive is **56**.

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