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 principle of inclusion-exclusion.
Here's how you can approach this problem step-by-step:
Step 1: Consider all possible ways of choosing 6 books out of 20 without any restrictions. This can be calculated using the combination formula. The number of ways to choose without any restrictions is denoted by C(n, r), where n is the total number of books (20 in this case) and r is the number of books you want to choose (6 in this case).
C(20, 6) = 20! / (6! * (20-6)!) = 38,760
Step 2: Count the number of ways to choose 6 books out of 20 where two adjacent books are selected. To do this, we can treat the two adjacent books as a single unit. So, we now have 19 objects (18 individual books + the unit of two adjacent books) instead of 20.
C(19, 6) = 19! / (6! * (19-6)!) = 27,132
Step 3: Subtract the count from Step 2 from the count from Step 1 to remove the cases where two adjacent books are selected.
38,760 - 27,132 = 11,628
Step 4: However, the count in Step 3 still includes the cases where three adjacent books are selected. So we need to add back those cases using a similar approach.
Step 5: Count the number of ways to choose 6 books out of 20 where three adjacent books are selected. Treat the three adjacent books as a single unit.
C(18, 6) = 18! / (6! * (18-6)!) = 18,564
Step 6: Add the count from Step 5 to the count in Step 3.
11,628 + 18,564 = 30,192
Step 7: However, the count in Step 6 includes the cases where four adjacent books are selected. So we need to subtract those cases.
Step 8: Count the number of ways to choose 6 books out of 20 where four adjacent books are selected. Treat the four adjacent books as a single unit.
C(17, 6) = 17! / (6! * (17-6)!) = 12,376
Step 9: Subtract the count from Step 8 from the count in Step 6.
30,192 - 12,376 = 17,816
Step 10: Repeat the process for each additional count where more adjacent books are selected. In this case, we don't have additional adjacent counts, so the final count is 17,816.
Therefore, there are 17,816 ways to choose 6 books out of 20 on a circular bookshelf such that no two adjacent books are selected.