Question

There are 9 bottles labelled 1, 2, ..., 9 and 9 boxes labelled 1, 2, ..., 9. The number of ways one can put these bottles in the boxes so that each box gets one bottle and exactly 5 bottles go in their corresponding numbered boxes is:

• A. \( 9 \times {}^9C_5 \)
• B. \( 5 \times {}^9C_5 \)
• C. \( 25 \times {}^9C_5 \)
• D. \( 44 \times {}^9C_5 \)

Hint:

The number of ways to choose \( k \) items to be in their correct positions out of \( n \) is \( \binom{n}{k} \). The remaining \( n-k \) items must be deranged.

Answer 1

The Correct Option is D

We need to find the number of permutations of 9 items where exactly 5 are in their correct positions. 
Step 1: Choose the 5 correctly placed bottles.
The number of ways to choose 5 bottles out of 9 to be placed in their corresponding boxes is given by the combination formula: \[ \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} = 126 \] Step 2: Consider the remaining 4 bottles and 4 boxes.
The remaining 4 bottles must be placed in the remaining 4 boxes such that none of them are in their corresponding numbered box. This is the number of derangements of 4 items, denoted by \( D_4 \) or \( !4 \). The formula for the number of derangements of \( n \) items is: \[ D_n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!} \] For \( n = 4 \): \[ D_4 = 4! \left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} \right) = 24 \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \right) = 24 \left( \frac{12 - 4 + 1}{24} \right) = 9 \] Step 3: Calculate the total number of ways.
The total number of ways to have exactly 5 bottles in their corresponding boxes is the product of the number of ways to choose the 5 correct bottles and the number of derangements of the remaining 4 bottles: \[ {Total ways} = \binom{9}{5} \times D_4 = 126 \times 9 = 1134 \] Now, we need to express this in the form \( k \times {}^9C_5 \). \[ k = D_4 = 9 \] There seems to be a discrepancy with the provided correct answer of \( 44 \times {}^9C_5 \). My calculation yields \( 9 \times {}^9C_5 \). There might be an error in the question or the provided options.