To solve this problem, we need to break the task down into two separate steps: choosing which 5 bottles go into their correct boxes, and then ensuring the remaining bottles go into completely incorrect boxes. The process of arranging objects such that none of them land in their correct position is known as a Derangement.
Step 1: Select the 5 bottles that go into their correct boxes
We have a total of 9 unique bottles and we need to choose exactly 5 of them to be placed in their matching correctly numbered boxes.
The number of ways to select 5 items out of 9 is given by the combination formula:
Ways to choose correct bottles=(59)
Step 2: Derange the remaining bottles
Once 5 bottles are placed correctly, we are left with:
9−5=4 bottles and 4 boxes
The problem states that exactly 5 bottles go into their correct boxes. This means that none of the remaining 4 bottles can go into their correct boxes. Therefore, we must perform a complete derangement of these 4 objects.
The standard formula for the number of derangements of n objects, denoted as Dn, is:
Dn=n!(2!1−3!1+4!1−⋯+n!(−1)n)
For n=4:
D4=4!(2!1−3!1+4!1)
Let us calculate the value inside the bracket:
D4=24(21−61+241)
Distribute 24 into the parenthesis:
D4=224−624+2424
D4=12−4+1=9
Step 3: Combine both steps for the total number of ways
According to the fundamental counting principle, the total number of valid distributions is the product of the combinations from Step 1 and the derangements from Step 2:
Total Ways=(59)×D4
Total Ways=9×(59)
Correct Answer
The correct option is (A) 9×(59).
