There are 9 bottles labeled 1, 2, 3, ..., 9 and 9 boxes labeled 1, 2, 3, ..., 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:
The problem involves placing 5 bottles in their corresponding boxes while the remaining 4 bottles do not go into their corresponding boxes (this is a derangement problem).
The solution process is as follows:
1. Place 5 bottles in their corresponding boxes:
The number of ways to place 5 bottles in their corresponding boxes is given by the combination formula \( \binom{9}{5} \).
2. Place the remaining 4 bottles in the boxes without matching their labels (Derangement):
The number of ways to arrange the 4 remaining bottles without placing any of them in their corresponding boxes (derangement) is 9.
3. Total ways:
The total number of ways to arrange the bottles is:
\[
\binom{9}{5} \times 9
\]
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