There are 10 different books on the shelf. Find the number of ways in which 3 books can be selected so that exactly two of them are consecutive.

To solve the problem of selecting 3 books from a shelf of 10, where exactly two of the selected books are consecutive, we can break it down into manageable steps. This involves understanding how to count combinations and applying some logical reasoning about the arrangement of the books.

Step-by-Step Breakdown

First, let’s clarify what it means for two books to be consecutive. If we have books labeled from 1 to 10, selecting books 1 and 2 means they are consecutive. Our goal is to select 3 books such that exactly two of them are next to each other, while the third book is not adjacent to these two.

Identifying Consecutive Pairs

We can think of the two consecutive books as a single unit or block. For example, if we select books 1 and 2 as our consecutive pair, we can represent them as a block (1, 2). This block can be placed in various positions on the shelf. Here’s how we can approach the counting:

• The consecutive pairs can be (1, 2), (2, 3), (3, 4), ..., (9, 10). This gives us a total of 9 possible pairs.

Choosing the Third Book

Once we have selected a pair, we need to choose a third book that is not adjacent to the pair. Let’s analyze the options based on the selected pair:

• If we select the pair (1, 2), the third book can be any of books 4 to 10 (7 options).
• If we select the pair (2, 3), the third book can be any of books 1 or 4 to 10 (8 options).
• If we select the pair (3, 4), the third book can be any of books 1, 2, or 5 to 10 (8 options).
• If we select the pair (4, 5), the third book can be any of books 1, 2, 3, or 6 to 10 (8 options).
• If we select the pair (5, 6), the third book can be any of books 1 to 4 or 7 to 10 (8 options).
• If we select the pair (6, 7), the third book can be any of books 1 to 5 or 8 to 10 (8 options).
• If we select the pair (7, 8), the third book can be any of books 1 to 6 or 9 to 10 (8 options).
• If we select the pair (8, 9), the third book can be any of books 1 to 7 or 10 (8 options).
• If we select the pair (9, 10), the third book can be any of books 1 to 8 (8 options).

Calculating Total Combinations

Now, let’s sum up the options for the third book based on the pairs:

• For the pair (1, 2): 7 options
• For the pair (2, 3): 8 options
• For the pair (3, 4): 8 options
• For the pair (4, 5): 8 options
• For the pair (5, 6): 8 options
• For the pair (6, 7): 8 options
• For the pair (7, 8): 8 options
• For the pair (8, 9): 8 options
• For the pair (9, 10): 8 options

Now, we can calculate the total number of ways:

Total = 7 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 7 + 8 * 8 = 7 + 64 = 71

Final Answer

Thus, the total number of ways to select 3 books from 10 such that exactly two of them are consecutive is 71.