Q6:What if the bookshelf is circular, that means you cannot choose the first and last books simultaneously? Count the number of ways to choose 6 books out of 20 books on a circular bookshelf such that no two adjacent books are selected.
To count the number of ways to choose 6 books out of 20 books on a circular bookshelf such that no two adjacent books are selected, we can use the concept of combinations.
Step 1: Calculate the total number of ways to choose 6 books out of 20 without any restrictions.
We can do this using the formula for combinations:
C(n, r) = n! / (r! * (n - r)!)
where n is the total number of books and r is the number of books to be chosen.
Using this formula, we can calculate:
C(20, 6) = 20! / (6! * (20 - 6)!) = 20! / (6! * 14!)
Step 2: Adjust for the restriction that no two adjacent books can be selected.
To account for this restriction, we need to subtract the number of ways we can choose 6 books with at least one pair of adjacent books.
We will first calculate the number of ways to choose 6 books with exactly one pair of adjacent books. In this case, we can treat the chosen pair as a single entity.
Step 2a: Calculate the number of ways to choose 6 books with exactly one pair of adjacent books.
We can fix the position of the chosen pair of adjacent books, resulting in a scenario where we need to choose 5 books out of the remaining 18 books (since we treat the pair as a single entity).
Using the same formula for combinations, we have:
C(18, 5) = 18! / (5! * (18 - 5)!)
Step 2b: Calculate the number of possible positions for the adjacent pair.
Since the chosen pair of adjacent books can be placed anywhere on the circular bookshelf, there are 20 possible positions.
Step 2c: Calculate the total number of ways to choose 6 books with at least one pair of adjacent books.
By multiplying the number of ways to choose 6 books with exactly one pair of adjacent books (Step 2a) by the number of possible positions for the adjacent pair (Step 2b), we get:
Total ways with at least one pair of adjacent books = C(18, 5) * 20
Step 3: Calculate the final number of ways to choose 6 books without any adjacent books.
To get the final number of ways to choose 6 books without any adjacent books, we subtract the total number of ways with at least one pair of adjacent books (Step 2c) from the total number of ways without any restrictions (Step 1):
Final number of ways = C(20, 6) - Total ways with at least one pair of adjacent books
Therefore, the number of ways to choose 6 books out of 20 books on a circular bookshelf such that no two adjacent books are selected is:
Final number of ways = (20! / (6! * 14!)) - (C(18, 5) * 20)
To count the number of ways to choose 6 books out of 20 on a circular bookshelf without selecting adjacent books, we can use the concept of inclusion-exclusion principle.
First, let's find the total number of ways to choose 6 books out of 20 without any restrictions. We can use the combination formula, which is given by:
C(n, r) = n! / (r!(n - r)!)
where n is the total number of books (20) and r is the number of books to be chosen (6).
C(20, 6) = 20! / (6!(20 - 6)!)
= 20! / (6!14!)
Next, let's count the number of ways to select adjacent books. Since the bookshelf is circular, we consider the cases where the selected books form a consecutive sequence.
To count the number of ways to choose 6 adjacent books, we can fix one book and then select 5 out of the remaining adjacent books. This can be calculated using the combination formula again.
C(19, 5) = 19! / (5!(19 - 5)!)
= 19! / (5!14!)
Since there are 20 books on the circular bookshelf, we have 20 different fixations. Therefore, the total number of ways to select 6 adjacent books is:
20 * C(19, 5)
Now, we need to exclude this count from the total count since we want to consider the cases where no two adjacent books are selected.
Finally, we can subtract the count of combinations where adjacent books are selected from the total count:
Total count - Count of combinations with adjacent books = C(20, 6) - (20 * C(19, 5))
By substituting the values into this equation, you can calculate the number of ways to choose 6 books out of 20 on a circular bookshelf such that no two adjacent books are selected.