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Question Id : 11627 | Context :NIMCET 2024

Question

There are 9 bottle labelled 1, 2, 3, ... , 9 and 9 boxes labelled 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 
🎥 Video solution / Text Solution of this question is given below:

Total bottles and boxes: 9 each, labeled 1 to 9.

We are asked to count permutations of bottles such that exactly 5 bottles go into their own numbered boxes.

Step 1: Choose 5 positions to be fixed points (i.e., bottle number matches box number).

Number of ways = \binom{9}{5}

Step 2: Remaining 4 positions must be a derangement (no bottle goes into its matching box).

Let D_4 be the number of derangements of 4 items.

D_4 = 9

Step 3: Total ways = \binom{9}{5} \times D_4 = 126 \times 9 = 1134

✅ Final Answer: \boxed{1134}


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