There are $9$ bottles and $9$ boxes numbered $1$ to $9$. The number of ways such that exactly $5$ bottles go to their corresponding boxes is
- A. $9\times{9 \choose 5}$
- B. $5\times{9 \choose 5}$
- C. $25\times{9 \choose 5}$
- D. $4\times{9 \choose 5}$
✅ Correct Answer: $9\times{9 \choose 5}$
Explanation
Choose $5$ bottles to be correctly placed Number of ways $={9 \choose 5}$ Remaining $4$ bottles must be deranged Number of derangements of $4$ objects is $9$ Total ways $=9\times{9 \choose 5}$
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